We now prove that $\gamma$ is attained. Let $\\{w_{j}\\}_{j}$ be a minimizing
sequence for $\gamma$. By the concentration-compactness principle,
$\\{w_{j}^{2}\,dx\\}_{j}$ is in the vanishing case if

$\lim_{j\to\infty}\int_{I_{R}}w_{j}^{2}=0\,,\qquad\forall R>0\,,$ (4.11)

where we have set $I_{R}=(-R,R)$. By (A.16) and (A.6) there exists $S_{0}$
such that

$W^{\prime\prime}(\eta)\geq\frac{1}{C}\,,\qquad\mbox{on $\mathbb{R}\setminus
I_{S_{0}}$}\,.$ (4.12)

Therefore by applying (4.11) twice with $R=S_{0}$ we find

$\displaystyle\limsup_{j\to\infty}\int_{\mathbb{R}}w_{j}^{2}$ $\displaystyle=$
$\displaystyle\limsup_{j\to\infty}\int_{\mathbb{R}\setminus
I_{S_{0}}}w_{j}^{2}\leq C\,\limsup_{j\to\infty}\int_{\mathbb{R}\setminus
I_{S_{0}}}W^{\prime\prime}(\eta)\,w_{j}^{2}$ $\displaystyle=$ $\displaystyle
C\,\limsup_{j\to\infty}\int_{\mathbb{R}}W^{\prime\prime}(\eta)\,w_{j}^{2}\leq\lim_{j\to\infty}Q(w_{j})=\gamma\leq
0$

a contradiction to $\int_{\mathbb{R}}w_{j}^{2}=1$. If, instead,
$\\{w_{j}^{2}\,dx\\}_{j}$ is in the dichotomy case, then there is
$\alpha\in(0,1)$ such that for every $\tau\in(0,\alpha/2)$ there exist $R>0$
and $R_{j}\to\infty$ as $j\to\infty$ such that

$\Big{|}1-\alpha-\int_{I_{R}}w_{j}^{2}\Big{|}<\tau\,,\qquad\Big{|}\alpha-\int_{\mathbb{R}\setminus
I_{R_{j}}}w_{j}^{2}\Big{|}<\tau\,,$ (4.13)

where, without loss of generality, we can assume $R\geq S_{0}$ for $S_{0}$ as
in (4.12). In particular, if $\varphi$ is a cut-off function between $I_{R}$
and $I_{R_{j}}$, then we have

$Q(w_{j})=Q(\varphi\,w_{j})+Q\big{(}(1-\varphi)\,w_{j}\big{)}+E_{j}\,,$ (4.14)

where, taking into account that $\varphi^{\prime}$ and $(1-\varphi)\,\varphi$
are supported in $I_{R_{j}}\setminus I_{R}$, we have

$\displaystyle E_{j}=2\,\int_{I_{R_{j}}\setminus
I_{R}}W^{\prime\prime}(\eta)\,(1-\varphi)\,\varphi\,w_{j}^{2}+4\,\int_{I_{R_{j}}\setminus
I_{R}}(\varphi w_{j})^{\prime}((1-\varphi)w_{j})^{\prime}\,.$ (4.15)

The first integral in (4.15) is non-negative by (4.12), while the second
integral contains a non-negative term of the form
$\varphi\,(1-\varphi)\,(w_{j}^{\prime})^{2}$: therefore, by (4.13),

$\displaystyle E_{j}$ $\displaystyle\geq$ $\displaystyle
4\,\int_{I_{R_{j}}\setminus
I_{R}}w_{j}\,\,w_{j}^{\prime}\,(1-\varphi)\,\varphi^{\prime}-w_{j}\,w_{j}^{\prime}\,\varphi\,\varphi^{\prime}-w_{j}^{2}\,(\varphi^{\prime})^{2}$
(4.16) $\displaystyle\geq$ $\displaystyle-C\,\int_{I_{R_{j}}\setminus
I_{R}}w_{j}^{2}-C\,\Big{(}\int_{I_{R_{j}}\setminus
I_{R}}w_{j}^{2}\Big{)}^{1/2}\,\Big{(}\int_{\mathbb{R}}(w_{j}^{\prime})^{2}\Big{)}^{1/2}\geq-C\,\sqrt{\tau}\,,$

where we have also used $Q(w_{j})\to\gamma$ as $j\to\infty$ to infer

$\int_{\mathbb{R}}(w_{j}^{\prime})^{2}\leq Q(w_{j})+\|W\|_{C^{0}[0,1]}\leq
C\,.$

We can take $\varphi$ supported in $I_{R+1}$. In this way, up to extracting a
subsequence, we have that $\varphi\,w_{j}$ admits a weak limit $w$ in
$H^{1}(\mathbb{R})$. By lower semicontinuity, homogeneity of $Q$ and (4.13) we
have that

$\liminf_{j\to\infty}Q(\varphi\,w_{j})\geq
Q(w)\geq\gamma\,\int_{\mathbb{R}}w^{2}\geq\,(1-\alpha)\,\gamma-C\,\tau\,.$
(4.17)

Finally, since $(1-\varphi)$ is supported on $\mathbb{R}\setminus I_{S_{0}}$,
by (4.12) we have

$\int_{\mathbb{R}}Q\big{(}(1-\varphi)\,v_{j}\big{)}\geq\frac{1}{C}\,\int_{\mathbb{R}}\,(1-\varphi)^{2}\,w_{j}^{2}\geq\frac{\alpha}{C}-C\,\tau\,,$

so that, combining (4.14), (4.16), and (4.17) we find

$\gamma\geq(1-\alpha)\,\gamma+\frac{\alpha}{C}-C\,\sqrt{\tau}\,.$

Letting $\tau\to 0^{+}$ we find a contradiction with $\gamma\leq 0$ and
$\alpha>0$. Having excluded vanishing and dichotomy, we have proved the
existence of minimizers of $\gamma$.

Let now $u$ be a minimizer of $\gamma$. Up to replace with $u$ with $|u|$ we
can assume $u\geq 0$. By a standard variational argument there exists
$\lambda\in\mathbb{R}$ such that

$\int_{\mathbb{R}}2u^{\prime}\,v^{\prime}+W^{\prime\prime}(\eta)\,u\,v=\lambda\int_{\mathbb{R}}u\,v\,,\qquad\forall
v\in H^{1}(\mathbb{R})\,.$ (4.18)

Testing with $v=\eta^{\prime}$ and recalling that
$2\,(\eta^{\prime})^{\prime\prime}=W^{\prime\prime}(\eta)\eta^{\prime}$, we
deduce that

$\lambda\int_{\mathbb{R}}\eta^{\prime}u=0,$

and, since $u\geq 0$, $\int_{\mathbb{R}}u^{2}=1$, and $\eta^{\prime}<0$, we
find $\lambda=0$. Thus $u$ is a $C^{2}$-solution of the ODE
$2\,u^{\prime\prime}=W^{\prime\prime}(\eta)\,u$ on $\mathbb{R}$. If
$u(r_{0})=0$ for some $r_{0}\in\mathbb{R}$, then $u^{\prime}(r_{0})\neq 0$
(otherwise we would have $u=0$ on $\mathbb{R}$, against
$\int_{\mathbb{R}}u^{2}=1$), and $u^{\prime}(r_{0})\neq 0$ contradicts $u\geq
0$ on $\mathbb{R}$. Hence, $u>0$ on $\mathbb{R}$.

Having proved that every minimizer of $\gamma$ is either positive or negative
on the whole $\mathbb{R}$, we deduce that $\gamma=0$. Indeed, if $u_{1}$ and
$u_{2}$ are minimizers of $\gamma$ and, say, they are both positive on
$\mathbb{R}$, then they solve (4.18) with $\lambda=0$, and thus

$2\,\gamma=Q(u_{1})+Q(u_{2})=Q(u_{1}-u_{2})\geq\gamma\,\int_{\mathbb{R}}(u_{1}-u_{2})^{2}=2\,\gamma\,\Big{(}1-\int_{\mathbb{R}}u_{1}\,u_{2}\Big{)}\,.$

Since $\int_{\mathbb{R}}u_{1}\,u_{2}\in(0,1)$, $\gamma<0$ would give a
contradiction. Having established that $\gamma=0$, we now know that
$\eta^{\prime}$ is a minimizer of $\gamma$. If $u$ is also minimizer of
$\gamma$, then, again by (4.18),

$Q(u+s\,\eta^{\prime})=Q(u)+s^{2}\,Q(\eta^{\prime})=0\qquad\forall
s\in\mathbb{R}\,.$

In particular, if $s\in\mathbb{R}$ is such that $u+s\,\eta^{\prime}$ is not
identically zero on $\mathbb{R}$, then
$(u+s\,\eta^{\prime})/\|u+s\,\eta^{\prime}\|_{L^{2}(\mathbb{R})}^{2}$ is a
minimizer of $\gamma$, and thus $u+s\,\eta^{\prime}$ is either positive or
negative on the whole $\mathbb{R}$. Let
$s_{0}=\inf\\{s:\mbox{$u+s\,\eta^{\prime}<0$ on $\mathbb{R}$}\\}$. If, say,
$u$ is a negative minimizer (like $\eta^{\prime}$ is), then $s_{0}\leq 0$;
while, clearly, $s_{0}>-\infty$, since, for $s$ negative enough, we must have
$u+s\,\eta^{\prime}>0$ at at least one point, and thus everywhere. Since
$u+s_{0}\,\eta^{\prime}\leq 0$ on $\mathbb{R}$ with $u+s_{0}\,\eta^{\prime}=0$
at at least one point, we deduce that $u+s_{0}\,\eta^{\prime}=0$ on
$\mathbb{R}$. ∎

######  Lemma 4.4 (Second stability lemma).

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.11) and (1.12), then there
exists a universal constant $\varepsilon_{0}$ with the following property. If
$u_{\varepsilon}\in\mathcal{R}_{0}^{*}$ is a minimizer of $\psi(\varepsilon)$
for $\varepsilon<\varepsilon_{0}$ and $h\in H^{1}(\mathbb{R}^{n})$ is a radial
function such that

$\int_{\mathbb{R}^{n}}V^{\prime}(u_{\varepsilon})\,h=0\,,$ (4.19)

then

$\int_{\mathbb{R}^{n}}2\,\varepsilon\,|\nabla
h|^{2}+\Big{(}\frac{W^{\prime\prime}(u_{\varepsilon})}{\varepsilon}-\lambda(\varepsilon)V^{\prime\prime}(u_{\varepsilon})\Big{)}\,h^{2}\geq\frac{1}{C}\,\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla
h|^{2}+\frac{h^{2}}{\varepsilon}\,,$ (4.20)

where $\lambda(\varepsilon)$ is the Lagrange multiplier of $u_{\varepsilon}$
as in (4.5).

###### Proof.

Step one: We show that is enough to prove the lemma with

$\int_{\mathbb{R}^{n}}2\,\varepsilon\,|\nabla
h|^{2}+\Big{(}\frac{W^{\prime\prime}(u_{\varepsilon})}{\varepsilon}-\lambda(\varepsilon)V^{\prime\prime}(u_{\varepsilon})\Big{)}\,h^{2}\geq\frac{1}{C}\,\int_{\mathbb{R}^{n}}\frac{h^{2}}{\varepsilon}\,,$
(4.21)

in place of (4.20). Indeed, if $\varepsilon_{0}$ is small enough, then
$|\lambda(\varepsilon)|\leq c(n)$ thanks to (2.3), and thus we can find a
universal constant $C_{*}$ such that

$\int_{\mathbb{R}^{n}}\Big{|}\frac{1}{\varepsilon}\,W^{\prime\prime}(u_{\varepsilon})-\lambda(\varepsilon)V^{\prime\prime}(u_{\varepsilon})\Big{|}\,h^{2}\leq
C_{*}\int_{\mathbb{R}^{n}}\frac{h^{2}}{\varepsilon}\,,$

whenever $u_{\varepsilon}$ is a minimizer of $\psi(\varepsilon)$,
$\varepsilon<\varepsilon_{0}$, and $h\in H^{1}(\mathbb{R}^{n})$. Let us now
fix a radial function $h\in H^{1}(\mathbb{R}^{n})$ satisfying (4.19). If
$C_{*}\,\int_{\mathbb{R}^{n}}h^{2}/\varepsilon\leq\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla
h|^{2}$, then we trivially have

$\displaystyle\int_{\mathbb{R}^{n}}2\,\varepsilon|\nabla
h|^{2}+\Big{(}\frac{W^{\prime\prime}(u_{\varepsilon})}{\varepsilon}-\lambda(\varepsilon)\,V^{\prime\prime}(\zeta_{\varepsilon})\Big{)}\,h^{2}\geq\int_{\mathbb{R}^{n}}2\,\varepsilon|\nabla
h|^{2}-C_{*}\,\int_{\mathbb{R}^{n}}\frac{h^{2}}{\varepsilon}\geq\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla
h|^{2}\,;$

if, instead,
$C_{*}\,\int_{\mathbb{R}^{n}}h^{2}/\varepsilon\geq\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla
h|^{2}$, then we deduce from (4.21)

$\int_{\mathbb{R}^{n}}2\,\varepsilon|\nabla
h|^{2}+\Big{(}\frac{W^{\prime\prime}(u_{\varepsilon})}{\varepsilon}-\lambda(\varepsilon)\,V^{\prime\prime}(u_{\varepsilon})\Big{)}\,h^{2}\geq\frac{1}{C}\,\int_{\mathbb{R}^{n}}\,\frac{h^{2}}{\varepsilon}\geq\frac{1}{C\,C_{*}}\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla
h|^{2}\,.$

In both cases, (4.20) is easily deduced.

Step two: We prove (4.21). We argue by contradiction, and consider
$\varepsilon_{j}\to 0^{+}$ as $j\to\infty$, $u_{j}\in\mathcal{R}_{0}^{*}$
minimizers of $\psi(\varepsilon_{j})$, and radial functions $h_{j}\in
H^{1}(\mathbb{R}^{n})$ such that

$\displaystyle\int_{\mathbb{R}^{n}}V^{\prime}(u_{j})\,h_{j}=0\,,$ (4.22)
$\displaystyle\int_{\mathbb{R}^{n}}2\,\varepsilon_{j}\,|\nabla
h_{j}|^{2}+\Big{(}\frac{W^{\prime\prime}(u_{j})}{\varepsilon_{j}}-\lambda_{j}\,V^{\prime\prime}(u_{j})\Big{)}\,h_{j}^{2}<\frac{1}{j}\,\int_{\mathbb{R}^{n}}\frac{h_{j}^{2}}{\varepsilon_{j}}\,.$
(4.23)

where $\lambda_{j}$ are the Lagrange multipliers corresponding to $u_{j}$. By
homogeneity of (4.22) and (4.23) we can also assume that

$\int_{\mathbb{R}^{n}}\frac{h_{j}^{2}}{\varepsilon_{j}}=1\,.$ (4.24)

Therefore, setting

$\eta_{j}(s)=u_{j}(R_{0}+\varepsilon_{j}\,s)\,,\qquad\beta_{j}(s)=h_{j}(R_{0}+\varepsilon_{j}\,s)\,,\qquad
s\geq-\frac{R_{0}}{\varepsilon_{j}}\,,$

we can recast (4.23) and (4.24) as

$\displaystyle\int_{-R_{0}/\varepsilon_{j}}^{\infty}\Big{(}2\,(\beta_{j}^{\prime})^{2}+(W^{\prime\prime}(\eta_{j})-\varepsilon_{j}\,\lambda_{j}\,V^{\prime\prime}(\eta_{j}))\,\beta_{j}^{2}\Big{)}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds\,\leq\frac{1}{j}\,,$
(4.26)
$\displaystyle\int_{-R_{0}/\varepsilon_{j}}^{\infty}\beta_{j}(s)^{2}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds=1\,.$

By $\varepsilon_{j}\to 0^{+}$ and by (2.3) we know $\lambda_{j}\to c(n)$ as
$j\to\infty$, which combined with $\|V^{\prime\prime}\|_{C^{0}[0,1]}\leq C$
and $\varepsilon_{j}\to 0^{+}$ shows that (4.26) and (4.26) imply

$\displaystyle\limsup_{j\to\infty}\int_{-R_{0}/\varepsilon_{j}}^{\infty}\Big{\\{}2\,(\beta_{j}^{\prime})^{2}+W^{\prime\prime}(\eta_{j})\,\beta_{j}^{2}\Big{\\}}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds\leq
0\,.$ (4.27)

Since $W^{\prime\prime}$ is bounded on $[0,1]$, by (4.26) and (4.27) we deduce
that $\\{\beta_{j}\\}_{j}$ is bounded in $H^{1}(-s_{0},s_{0})$ for every
$s_{0}>0$. In particular there exists $\beta\in H^{1}_{{\rm loc}}(\mathbb{R})$
such that, up to extracting subsequences, $\beta$ is the weak limit of
$\\{\beta_{j}\\}_{j}$ in $H^{1}(-s_{0},s_{0})$ for every $s_{0}>0$. By
$\beta_{j}^{\prime}\rightharpoonup\beta^{\prime}$ in $L^{2}(-s_{0},s_{0})$ for
every $s_{0}>0$ we easily find

$\displaystyle\liminf_{j\to\infty}\int_{-R_{0}/\varepsilon_{j}}^{\infty}2\,\beta_{j}^{\prime}(s)^{2}(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds\geq
R_{0}^{n-1}\,\int_{\mathbb{R}}2\,(\beta^{\prime})^{2}\,.$ (4.28)

We now apply the concentration-compactness principle to the sequence of
measures

$\mu_{j}=1_{(-R_{0}/\varepsilon_{j},\infty)}(s)\,\beta_{j}(s)^{2}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds\,,$

which satisfy $\mu_{j}(\mathbb{R})=1$ thanks to (4.24). We claim that, if the
compactness case hold, and thus

$\lim_{s_{0}\to+\infty}\sup_{j}\,\mu_{j}(\mathbb{R}\setminus[-s_{0},s_{0}])=0\,,$
(4.29)

then we can reach a contradiction, and complete the proof of the lemma. To
prove this claim, let us set

$\eta_{0}(s)=\eta(s-\tau_{0})\,,$

for $\tau_{0}$ as in (A.19), and let us notice that, for every $s_{0}>0$ we
have

$\displaystyle\limsup_{j\to\infty}\Big{|}\int_{-R_{0}/\varepsilon_{j}}^{\infty}W^{\prime\prime}(\eta_{j})\beta_{j}(s)^{2}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds-
R_{0}^{n-1}\,\int_{\mathbb{R}}W^{\prime\prime}(\eta_{0})\,\beta^{2}\Big{|}$
$\displaystyle\leq$
$\displaystyle\limsup_{j\to\infty}\int_{-s_{0}}^{s_{0}}\Big{|}W^{\prime\prime}(\eta_{j})\beta_{j}(s)^{2}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}-R_{0}^{n-1}\,W^{\prime\prime}(\eta_{0})\,\beta^{2}\Big{|}$
$\displaystyle+\|W^{\prime\prime}\|_{C^{0}[0,1]}\,\sup_{j\in\mathbb{N}}\mu_{j}(\mathbb{R}\setminus[-s_{0},s_{0}])+R_{0}^{n-1}\,\|W^{\prime\prime}\|_{C^{0}[0,1]}\,\int_{\mathbb{R}\setminus[-s_{0},s_{0}]}\beta^{2}\,.$

Since $\beta_{j}\to\beta$ in $L^{2}_{{\rm loc}}(\mathbb{R})$ and
$\eta_{j}\to\eta_{0}$ locally uniformly on $\mathbb{R}$ thanks to Theorem 3.1,
the first term on the right-hand side of (4) is equal to zero. Letting now
$s_{0}\to\infty$, the second term goes to zero thanks to (4.29), while the
third terms goes to zero thanks to the fact that (4.29) implies in particular

$R_{0}^{n-1}\,\int_{\mathbb{R}}\beta^{2}=1\,.$ (4.31)

We can combine this information with (4.28) and finally deduce from (4.27)
that

$\int_{\mathbb{R}}\,2\,(\beta^{\prime})^{2}+W^{\prime\prime}(\eta_{0})\,\beta^{2}\leq
0\,.$ (4.32)

By Lemma 4.3 we deduce that, if we set $\beta_{0}(s)=\beta(s+\tau_{0})$, then
$\beta_{0}=t\,\eta^{\prime}$ for some $t\neq 0$ ($t=0$ being ruled out by
(4.31)). In particular, $\beta=t\,\eta_{0}^{\prime}$, and therefore

$\int_{\mathbb{R}}\,V^{\prime}(\eta_{0})\,\beta=t\,V(\eta_{0})|^{+\infty}_{-\infty}=t\,V(1)=t\neq
0\,.$

However, by (4.22), we see that

$0=\int_{\mathbb{R}^{n}}V^{\prime}(u_{j})\,h_{j}=\int_{-R_{0}/\varepsilon_{j}}^{\infty}\,V^{\prime}(\eta_{j})\,\beta_{j}(s)\,(R_{0}+s\,\varepsilon_{j})^{n-1}\,ds\,,\qquad\forall
j\,,$

and we can thus obtain a contradiction by showing that

$\lim_{j\to\infty}\int_{-R_{0}/\varepsilon_{j}}^{\infty}\,V^{\prime}(\eta_{j})\,\beta_{j}(s)\,(R_{0}+s\,\varepsilon_{j})^{n-1}\,ds=R_{0}^{n-1}\,\int_{\mathbb{R}}\,V^{\prime}(\eta_{0})\,\beta\,.$
(4.33)

This is proved by noticing that (A.11), (A.16), (3.56) and (3.57) give

$0\leq\max\\{V^{\prime}(\eta_{j}),V^{\prime}(\eta_{0})\\}\leq
C\,e^{-|s|/C}\,,$

for every $s\in\mathbb{R}$ (or for every $s\geq-R_{0}/\varepsilon_{j}$, in the
case of $\eta_{j}$). In particular,

$\displaystyle\lim_{s_{0}\to\infty}\limsup_{j\to\infty}\,\Big{[}\int_{-R_{0}/\varepsilon_{j}}^{-s_{0}}+\int_{s_{0}}^{\infty}\Big{]}V^{\prime}(\eta_{j})\,|\beta_{j}|\,(R_{0}+s\,\varepsilon_{j})^{n-1}\,ds$
$\displaystyle\leq\,C\,\lim_{s_{0}\to\infty}\,\limsup_{j\to\infty}\Big{(}\int_{\\{|s|>s_{0}\\}}e^{-|s|/C}\,(R_{0}+s\,\varepsilon_{j})^{n-1}\,ds\Big{)}^{1/2}\,\mu_{j}(\mathbb{R}\setminus[-s_{0},s_{0}])^{1/2}=0\,,$

so that a similar argument to the one used in (4) can be repeated to prove
(4.33).

We are thus left to prove that the sequence of probability measures
$\\{\mu_{j}\\}_{j}$ cannot be in the vanishing case nor in the dichotomy case
of the concentration-compactness principle.

To exclude that $\\{\mu_{j}\\}_{j}$ is in the vanishing case: Since
$\eta_{j}\to\eta$ locally uniformly on $\mathbb{R}$, up to take $j$ large
enough and for $S_{0}$ as in (4.12) we have $W^{\prime\prime}(\eta_{j}(s))\geq
1/C$ for $|s|\geq S_{0}$, $s\geq-R_{0}/\varepsilon_{j}$. Since we are in the
vanishing case, it holds

$\lim_{j\to\infty}\int_{-S_{0}}^{S_{0}}\,\beta_{j}(s)^{2}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds=0\,,$
(4.34)

so that, by using first the lower bound on $W^{\prime\prime}$, and then
(4.34), we get

$\displaystyle\frac{1}{C}\,\limsup_{j\to\infty}\Big{[}\int_{-R_{0}/\varepsilon_{j}}^{-S_{0}}+\int_{S_{0}}^{\infty}\Big{]}\beta_{j}(s)^{2}\,\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds$
$\displaystyle\leq$
$\displaystyle\limsup_{j\to\infty}\Big{[}\int_{-R_{0}/\varepsilon_{j}}^{-S_{0}}+\int_{S_{0}}^{\infty}\Big{]}W^{\prime\prime}(\eta_{j})\,\beta_{j}(s)^{2}\,\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds$
$\displaystyle=$
$\displaystyle\limsup_{j\to\infty}\int_{-R_{0}/\varepsilon_{j}}^{\infty}\,W^{\prime\prime}(\eta_{j})\,\beta_{j}(s)^{2}\,\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds\leq
0$

where in the last inequality we have used (4.27). Combining this information
with (4.34) we obtain a contradiction to (4.26), thus excluding the vanishing
case.

To exclude that $\\{\mu_{j}\\}_{j}$ is in the dichotomy case: With $S_{0}$ as
above, if we are in the dichotomy case, then there exists $\alpha\in(0,1)$
such that for every $\tau\in(0,\alpha/2)$ there exist $R>S_{0}$ and
$R_{j}\to\infty$ such that

$|\mu_{j}(I_{R})-(1-\alpha)|<\tau\,,\qquad|\mu_{j}(\mathbb{R}\setminus
I_{R_{j}})-\alpha|<\tau\,,\qquad\forall j\,.$ (4.35)

Setting $A_{j}=\varphi\,\beta_{j}$, $B_{j}=(1-\varphi)\,\beta_{j}$, where
$\varphi$ is a cut-off function between $B_{R}$ and $B_{R+1}$, and setting for
the sake of brevity,

$Q_{j}(A,B)=\int_{-R_{0}/\varepsilon_{j}}^{\infty}\Big{\\{}2\,A^{\prime}\,B^{\prime}+W^{\prime\prime}(\eta_{j})\,A\,B\Big{\\}}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds\,,\qquad
Q_{j}(A)=Q_{j}(A,A)\,,$

we can rewrite (4.27) as

$\limsup_{j\to\infty}Q_{j}(A_{j})+Q_{j}(B_{j})+2\,Q_{j}(A_{j},B_{j})\leq 0\,,$
(4.36)

Now, since $\varphi^{\prime}$ and $(1-\varphi)\,\varphi$ are supported in
$I_{R+1}\setminus I_{R}$, we see that

$\displaystyle Q_{j}(A_{j},B_{j})$ $\displaystyle\geq$ $\displaystyle
2\,\int_{I_{R+1}\setminus
I_{R}}\,(1-2\,\varphi)\,\varphi^{\prime}\,\beta_{j}\,\beta_{j}^{\prime}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds$
$\displaystyle+\int_{I_{R+1}\setminus
I_{R}}\,\Big{\\{}W^{\prime\prime}(\eta_{j})-(\varphi^{\prime})^{2}\Big{\\}}\,\beta_{j}^{2}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds$

where, thanks to (4.27) and Hölder inequality,

$\displaystyle\int_{I_{R+1}\setminus
I_{R}}\,(1-2\,\varphi)\,\varphi^{\prime}\,\beta_{j}\,\beta_{j}^{\prime}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds\leq
C\,\mu_{j}(I_{R+1}\setminus I_{R})^{1/2}\leq C\,\sqrt{\tau}\,$
$\displaystyle\int_{I_{R+1}\setminus
I_{R}}\,\Big{\\{}W^{\prime\prime}(\eta_{j})-(\varphi^{\prime})^{2}\Big{\\}}\,\beta_{j}^{2}\,(R_{0}+\varepsilon_{j}\,s)^{n-1}\,ds\leq
C\,\mu_{j}(I_{R+1}\setminus I_{R})\leq C\,\tau\,.$

We thus conclude that $Q_{j}(A_{j},B_{j})\geq-C\,\sqrt{\tau}$ for every $j$,
and thus, by (4.36), that

$\limsup_{j\to\infty}Q_{j}(A_{j})+Q_{j}(B_{j})\leq C\,\sqrt{\tau}\,.$ (4.37)

Now, since the supports of the $A_{j}$’s are uniformly bounded, we easily see
that there exists $A\in H^{1}(\mathbb{R})$ such that $A_{j}\rightharpoonup A$
weakly in $H^{1}(\mathbb{R})$; in particular,

$\liminf_{j\to\infty}Q_{j}(A_{j})\geq\int_{\mathbb{R}}\,2\,(A^{\prime})^{2}+W^{\prime\prime}(\eta_{0})\,A^{2}\geq
0\,,$

where in the last inequality we have used Lemma 4.3. By combining this last
inequality with (4.37), $W^{\prime\prime}(\eta_{j})\geq 1/C$ on
$\mathbb{R}\setminus I_{S_{0}}$, and $R\geq S_{0}$, we conclude that

$C\,\sqrt{\tau}\geq\limsup_{j\to\infty}Q_{j}(B_{j})\geq\frac{1}{C}\,\limsup_{j\to\infty}\int_{-R_{0}/\varepsilon_{j}}^{\infty}\,(1-\varphi)^{2}\,\beta_{j}^{2}\,(R+s\,\varepsilon_{j})^{n-1}\,ds$

and thus, by (4.35), that $C\,\sqrt{\tau}\geq(\alpha/C)-C\,\tau$. Letting
$\tau\to 0^{+}$ we obtain a contradiction with $\alpha>0$. ∎

######  Lemma 4.5 (Third stability lemma).

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.11) and (1.12), then there
exist universal constants $\delta_{0}$ and $\varepsilon_{0}$ such that, if
$u_{\varepsilon}\in\mathcal{R}_{0}^{*}$ is a minimizer of $\psi(\varepsilon)$
for $\varepsilon<\varepsilon_{0}$ and $u\in H^{1}(\mathbb{R}^{n};[0,1])$ is a
radial function with

$\displaystyle\int_{\mathbb{R}^{n}}V(u)=1\,,$ (4.38)
$\displaystyle\int_{\mathbb{R}^{n}}(u-u_{\varepsilon})^{2}\leq
C\,\varepsilon\,,$ (4.39)
$\displaystyle\|u-u_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{n})}\leq\delta_{0}\,,$
(4.40)

then, setting $h=u-u_{\varepsilon}$,

$\int_{\mathbb{R}^{n}}2\,\varepsilon\,|\nabla
h|^{2}+\Big{(}\frac{W^{\prime\prime}(u_{\varepsilon})}{\varepsilon}-\lambda(\varepsilon)V^{\prime\prime}(u_{\varepsilon})\Big{)}\,h^{2}\geq\frac{1}{C}\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla
h|^{2}+\frac{h^{2}}{\varepsilon}\,.$ (4.41)

where $\lambda(\varepsilon)$ is the Lagrange multiplier of $u_{\varepsilon}$
as in (4.5).

###### Proof.

It will be convenient to set

$\displaystyle P_{\varepsilon}(u,v)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\varepsilon\,\nabla u\cdot\nabla
v+\frac{u\,v}{\varepsilon}\,,$ $\displaystyle Q_{\varepsilon}(u,v)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{n}}\varepsilon\,\nabla
u\cdot\nabla
v+\Big{(}\frac{W^{\prime\prime}(u_{\varepsilon})}{\varepsilon}-\lambda(\varepsilon)\,V^{\prime\prime}(u_{\varepsilon})\Big{)}\,u\,v\,,$

as well as $P_{\varepsilon}(u)=P_{\varepsilon}(u,u)$ and
$Q_{\varepsilon}(u)=Q_{\varepsilon}(u,u)$. Let us start noticing that by
Theorem 3.1 we have

$\lim_{\sigma\to
0}\,\sup_{\varepsilon<\sigma}\,\sup_{v_{\varepsilon}}\,\Big{|}\int_{\mathbb{R}^{n}}V^{\prime}(v_{\varepsilon})\,v_{\varepsilon}-R_{0}^{n-1}\,\int_{\mathbb{R}}V^{\prime}(\eta)\,\eta\Big{|}=0\,,$

where $v_{\varepsilon}$ runs over all radial minimizers of
$\psi(\varepsilon)$. Since $\int_{\mathbb{R}}V^{\prime}(\eta)\,\eta$ is a
positive constant depending on $n$ and $W$ only this shows in particular that

$\frac{1}{C}\leq\int_{\mathbb{R}^{n}}V^{\prime}(u_{\varepsilon})\,u_{\varepsilon}\leq
C\,,\qquad\forall\varepsilon<\varepsilon_{0}\,.$ (4.42)

By (4.42), given $h=u-u_{\varepsilon}$ as in the statement, we can always find
$t\in\mathbb{R}$ such that

$\int_{\mathbb{R}^{n}}V^{\prime}(u_{\varepsilon})\,(h+t\,u_{\varepsilon})=0\,,\qquad\mbox{i.e.}\quad
t=-\frac{\int_{\mathbb{R}^{n}}V^{\prime}(u_{\varepsilon})\,h}{\int_{\mathbb{R}^{n}}V^{\prime}(u_{\varepsilon})\,u_{\varepsilon}}\,.$
(4.43)

By (A.12), (4.40), and since $0\leq u_{\varepsilon}+h\leq 1$, we have that, on
$\mathbb{R}^{n}$,

$\Big{|}V(u_{\varepsilon}+h)-V(u_{\varepsilon})-V^{\prime}(u_{\varepsilon})\,h-V^{\prime\prime}(u_{\varepsilon})\,\frac{h^{2}}{2}\Big{|}\leq
C\,\delta_{0}\,h^{2}\,,$ (4.44)

so that, by (4.38),

$\displaystyle\Big{|}\int_{\mathbb{R}^{n}}V^{\prime}(u_{\varepsilon})\,h+V^{\prime\prime}(u_{\varepsilon})\,\frac{h^{2}}{2}\Big{|}\leq
C\,\delta_{0}\,\int_{\mathbb{R}^{n}}h^{2}\,,$ (4.45)

and thus, thanks to $\|V^{\prime\prime}\|_{C^{0}[0,1]}\leq C$, (4.42), (4.39),
and (4.43),

$|t|\leq C\,\int_{\mathbb{R}^{n}}h^{2}\leq
C\,\varepsilon\,\min\\{P_{\varepsilon}(h),1\\}\,,$ (4.46)

By (4.43) we can apply Lemma 4.4 to $u_{\varepsilon}+t\,h$ and find that

$Q_{\varepsilon}(h+t\,u_{\varepsilon})\geq\frac{P_{\varepsilon}(h+t\,u_{\varepsilon})}{C}\,,$

which can be more conveniently rewritten as

$Q_{\varepsilon}(h)\geq\frac{P_{\varepsilon}(h)}{C}+2\,t\,\Big{\\{}\frac{P_{\varepsilon}(h,u_{\varepsilon})}{C}-Q_{\varepsilon}(h,u_{\varepsilon})\Big{\\}}+t^{2}\,\Big{\\{}\frac{P_{\varepsilon}(u_{\varepsilon})}{C}-Q_{\varepsilon}(u_{\varepsilon})\Big{\\}}\,.$
(4.47)

By Theorem 3.1, we see that
$P_{\varepsilon}(u_{\varepsilon})+|Q_{\varepsilon}(u_{\varepsilon})|\leq C$
(uniformly on $\varepsilon<\varepsilon_{0}$), so that (4.47) and (4.46) give

$Q_{\varepsilon}(h)\geq\frac{P_{\varepsilon}(h)}{C}+2\,t\,\Big{\\{}\frac{P_{\varepsilon}(h,u_{\varepsilon})}{C}-Q_{\varepsilon}(h,u_{\varepsilon})\Big{\\}}\,.$
(4.48)

By Hölder inequality, $ab\leq(a^{2}+b^{2})/2$,
$P_{\varepsilon}(u_{\varepsilon})\leq C$, and (4.46) we see that

$|t|\,P_{\varepsilon}(h,u_{\varepsilon})\leq\frac{|t|}{2}\,\big{(}P_{\varepsilon}(h)+P_{\varepsilon}(u_{\varepsilon})\big{)}\leq
C\,\varepsilon\,P_{\varepsilon}(h)\,,$ (4.49)

while by $|V^{\prime}|+|W^{\prime\prime}|\leq C$ and
$|\lambda(\varepsilon)|\leq C$ for $\varepsilon<\varepsilon_{0}$ we find,
arguing as in (4.49),

$|t|\,Q_{\varepsilon}(h,u_{\varepsilon})\leq|t|\,\Big{\\{}\varepsilon\,\int_{\mathbb{R}^{n}}|\nabla
h|\,|\nabla
u_{\varepsilon}|+\frac{C}{\varepsilon}\,\int_{\mathbb{R}^{n}}|h|\,u_{\varepsilon}\Big{\\}}\leq
C\,\varepsilon\,P_{\varepsilon}(h)\,.$ (4.50)

By combining (4.48), (4.49), and (4.50) we conclude that
$Q_{\varepsilon}(h)\geq P_{\varepsilon}(h)/C$, as desired. ∎

We are finally ready to prove Theorem 4.1.

###### Proof of Theorem 4.1.

We are given $u_{\varepsilon}$ and $h$ as in Lemma 4.5, and now want to prove
that

$\mathcal{AC}_{\varepsilon}(u_{\varepsilon}+h)-\psi(\varepsilon)\geq\frac{1}{C}\,\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla
h|^{2}+\frac{h^{2}}{\varepsilon}\,,$ (4.51)

holds. By (A.5) and (4.40) we have that

$\Big{|}W(u_{\varepsilon}+h)-W(u_{\varepsilon})-W^{\prime}(u_{\varepsilon})\,h-W^{\prime\prime}(u_{\varepsilon})\,\frac{h^{2}}{2}\Big{|}\leq
C\,\delta_{0}\,h^{2}\,,\qquad\mbox{on $\mathbb{R}^{n}$}\,,$

therefore

$\displaystyle\mathcal{AC}_{\varepsilon}(u_{\varepsilon}+h)-\mathcal{AC}_{\varepsilon}(u_{\varepsilon})$
$\displaystyle\geq$ $\displaystyle\int_{\mathbb{R}^{n}}2\,\varepsilon\,\nabla
u_{\varepsilon}\cdot\nabla
h+\frac{W^{\prime}(u_{\varepsilon})}{\varepsilon}\,h$ (4.52)
$\displaystyle+\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla
h|^{2}+\frac{W^{\prime\prime}(u_{\varepsilon})}{2\,\varepsilon}\,h^{2}-C\,\delta_{0}\,\int_{\mathbb{R}^{n}}h^{2}\,.$

By the Euler–Lagrange equation for $u_{\varepsilon}$, see (2.1), we have

$\int_{\mathbb{R}^{n}}2\,\varepsilon\,\nabla u_{\varepsilon}\cdot\nabla
h+\frac{W^{\prime}(u_{\varepsilon})}{\varepsilon}\,h=\lambda(\varepsilon)\,\int_{\mathbb{R}^{n}}V^{\prime}(u_{\varepsilon})\,h\,.$
(4.53)

Moreover, by (4.45),

$\Big{|}\int_{\mathbb{R}^{n}}V^{\prime}(u_{\varepsilon})h+\int_{\mathbb{R}^{n}}V^{\prime\prime}(u_{\varepsilon})\frac{h^{2}}{2}\Big{|}\leq
C\,\delta_{0}\,\,\int_{\mathbb{R}^{n}}h^{2}\,.$ (4.54)

On combining (4.52), (4.53), and (4.54) with (4.41) we find that

$\displaystyle\mathcal{AC}_{\varepsilon}(u_{\varepsilon}+h)-\psi(\varepsilon)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{n}}2\,\varepsilon\,|\nabla
h|^{2}+\Big{\\{}\frac{1}{\varepsilon}\,W^{\prime\prime}(u_{\varepsilon})-\lambda(\varepsilon)\,V^{\prime\prime}(u_{\varepsilon})\Big{\\}}\,h^{2}-C\,\delta_{0}\,\int_{\mathbb{R}^{n}}h^{2}$
$\displaystyle\geq$ $\displaystyle\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla
h|^{2}+\frac{h^{2}}{\varepsilon}-C\,\delta_{0}\,\int_{\mathbb{R}^{n}}h^{2}\,,$

so that (4.51) follows by taking $\delta_{0}$ small enough. ∎

##  5\. Proof of the uniform stability theorem

In this section we prove Theorem 1.1-(iii), i.e., we prove (1.21). We focus
directly on the case $(\sigma,m)=(\varepsilon,1)$, from which the general case
follows immediately by scaling.

######  Theorem 5.1.

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.11) and (1.12), then there
exist universal constants $\varepsilon_{0}>0$ and $C$ such that, if
$\varepsilon<\varepsilon_{0}$ and $u\in H^{1}(\mathbb{R}^{n};[0,1])$ with
$\int_{\mathbb{R}^{n}}V(u)=1$, then

$C\,\sqrt{\mathcal{AC}_{\varepsilon}(u)-\psi(\varepsilon)}\geq\inf_{x_{0}\in\mathbb{R}^{n}}\,\int_{\mathbb{R}^{n}}\big{|}\Phi(u)-\Phi(T_{x_{0}}u_{\varepsilon})\big{|}^{n/(n-1)}$
(5.1)

where $T_{x_{0}}u_{\varepsilon}(x)=u_{\varepsilon}(x-x_{0})$,
$x\in\mathbb{R}^{n}$, and $u_{\varepsilon}$ denotes the unique minimizer of
$\psi(\varepsilon)$ in $\mathcal{R}_{0}$.

In order to streamline the exposition of the proof of Theorem 5.1, we
introduce the isoperimetric deficit and asymmetry of $u\in
H^{1}(\mathbb{R}^{n};[0,1])$ with $\int_{\mathbb{R}^{n}}V(u)=1$, by setting

$\displaystyle\delta_{\varepsilon}(u)$ $\displaystyle=$
$\displaystyle\mathcal{AC}_{\varepsilon}(u)-\psi(\varepsilon)\,,$
$\displaystyle\alpha_{\varepsilon}(u)$ $\displaystyle=$
$\displaystyle\inf_{x_{0}\in\mathbb{R}^{n}}d_{\Phi}(u,T_{x_{0}}u_{\varepsilon})\,.$

Here, as in Theorem 2.2,

$d_{\Phi}(u,v)=\int_{\mathbb{R}^{n}}|\Phi(u)-\Phi(v)|^{n/(n-1)}\,,\qquad\forall
u,v\in H^{1}(\mathbb{R}^{n};[0,1])\,.$

With this notation, Theorem 5.1 states the existence of universal constants
$C$ and $\varepsilon_{0}$ such that, if $\varepsilon<\varepsilon_{0}$, then

$C\,\sqrt{\delta_{\varepsilon}(u)}\geq\alpha_{\varepsilon}(u)\,,\qquad\forall
u\in H^{1}(\mathbb{R}^{n};[0,1])\,,\int_{\mathbb{R}^{n}}V(u)=1\,.$ (5.2)

In the following subsections we discuss some key steps of the proof of Theorem
5.1, which is then presented at the end of this section.

###  5.1. Reduction to the small asymmetry case

Thanks to the volume constraint $\int_{\mathbb{R}^{n}}V(u)=1$ and to the
triangular inequality in $L^{n/(n-1)}$, we always have
$\alpha_{\varepsilon}(u)\leq 2^{n/(n-1)}$. In particular, in proving (5.2), we
can always assume that $\delta_{\varepsilon}(u)\leq\delta_{0}$ for a universal
constant $\delta_{0}$. This is useful because, by the following lemma, by
assuming $\delta_{\varepsilon}(u)\leq\delta_{0}$ we can take
$\alpha_{\varepsilon}(u)$ as small as needed in dependence of $n$ and $W$.

######  Lemma 5.2 ($\varepsilon$-uniform qualitative stability).

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.11) and (1.12), then there
exists a universal constant $\varepsilon_{0}$ with the following property: for
every $\alpha>0$ there exists $\delta>0$ such that

$u\in
H^{1}(\mathbb{R}^{n};[0,1])\,,\qquad\int_{\mathbb{R}^{n}}V(u)=1\,,\qquad\varepsilon<\varepsilon_{0}\,,\qquad\delta_{\varepsilon}(u)\leq\delta\,,$

imply

$\alpha_{\varepsilon}(u)\leq\alpha\,.$

###### Proof.

We pick $\varepsilon_{0}$ such that Theorem 2.1 and Corollary 4.2 hold. If the
lemma is false for such $\varepsilon_{0}$, then there exists $\alpha_{*}>0$
and a sequence $\\{u_{j}\\}_{j}$ in $H^{1}(\mathbb{R}^{n};[0,1])$ with
$\int_{\mathbb{R}^{n}}V(u_{j})=1$ such that

$\delta_{\varepsilon_{j}}(u_{j})\to 0^{+}\qquad\mbox{as $j\to\infty$}\,,$
(5.3)

for some $\varepsilon_{j}\to\varepsilon_{*}\in[0,\varepsilon_{0}]$ and with
$\alpha_{\varepsilon_{j}}(u_{j})\geq\alpha_{*}$. By (5.3), there is
$\ell_{j}\to 0^{+}$ as $j\to\infty$ such that

$\mathcal{AC}_{\varepsilon_{j}}(u_{j})\leq\psi(\varepsilon_{j})+\ell_{j}\,,\qquad\forall
j\,,$ (5.4)

We now distinguish two cases:

Case one, $\varepsilon_{*}>0$: In this case, by continuity of $\psi$ (see
Theorem 2.1) and since

$\mathcal{AC}_{\varepsilon_{*}}(u_{j})-\psi(\varepsilon_{*})\leq
b_{j}\,\big{(}\mathcal{AC}_{\varepsilon_{j}}(u_{j})-\psi(\varepsilon_{j})\big{)}+b_{j}\,\psi(\varepsilon_{j})-\psi(\varepsilon_{*})\,,\qquad
b_{j}=\max\Big{\\{}\frac{\varepsilon_{j}}{\varepsilon_{*}},\frac{\varepsilon_{*}}{\varepsilon_{j}}\Big{\\}}\,,$

we can assume that
$\mathcal{AC}_{\varepsilon_{*}}(u_{j})-\psi(\varepsilon_{*})\leq\ell_{0}$ for
$\ell_{0}$ as in step two of the proof of Theorem 2.1. We can thus apply that
statement and conclude that, up to translations and up to subsequences, there
is $u\in H^{1}(\mathbb{R}^{n};[0,1])$ with $\int_{\mathbb{R}^{n}}V(u)=1$ such
that $d_{\Phi}(u_{j},u)\to 0$ as $j\to\infty$. In particular, $u$ is a
minimizer of $\psi(\varepsilon_{*})$, and therefore, up to a translation, we
can assume that $u=u_{\varepsilon_{*}}\in\mathcal{R}_{0}$. Now, by repeating
this same argument with the minimizers $u_{\varepsilon_{j}}$ of
$\psi(\varepsilon_{j})$ in $\mathcal{R}_{0}$ in place of $u_{j}$, we see that

$d_{\Phi}(u_{\varepsilon_{j}},u_{\varepsilon_{*}})\to 0\qquad\mbox{as
$j\to\infty$}\,,$

so that, thanks to (2.59), we find the contradiction

$\alpha_{*}\leq\alpha_{\varepsilon_{j}}(u_{j})\leq
d_{\Phi}(u_{j},u_{\varepsilon_{j}})\leq
d_{\Phi}(u_{j},u_{\varepsilon_{*}})+C\,d_{\Phi}(u_{\varepsilon_{j}},u_{\varepsilon_{*}})^{(n-1)/n}\to
0^{+}\,,$

as $j\to\infty$.

Case two, $\varepsilon_{*}=0$: In this case, thanks to (5.4),

$2\,|D[\Phi(u_{j})]|(\mathbb{R}^{n})\leq\mathcal{AC}_{\varepsilon_{j}}(u_{j})\leq\psi(\varepsilon_{j})+\ell_{j}\leq
2\,n\,\omega_{n}^{1/n}+C\,\varepsilon_{j}+\ell_{j}\,,$

so that $\\{\Phi(u_{j})\\}_{j}$ is asymptotically optimal for the sharp
$BV$-Sobolev inequality. By the concentration-compactness principle (see,
e.g., [FMP07, Theorem A.1]), up to subsequences and up to translations,
$\Phi(u_{j})\to a\,1_{B_{r}}$ in $L^{n/(n-1)}(\mathbb{R}^{n})$ as
$j\to\infty$, for some $a$ and $r$ such that
$a^{n/(n-1)}\,\omega_{n}\,r^{n}=1$. The fact that
$\mathcal{AC}_{\varepsilon_{j}}(v_{j})$ is bounded implies that
$v_{j}\to\\{0,1\\}$ a.e. on $\mathbb{R}^{n}$, therefore, by $\Phi(0)=0$ and
$\Phi(1)=1$, it must be $a=1$ and $R=R_{0}$ for $\omega_{n}\,R_{0}^{n}=1$. By
Theorem 3.1, if $u_{\varepsilon_{j}}$ is a the minimizer of
$\psi(\varepsilon_{j})$ in $\mathcal{R}_{0}$, then

$d_{\Phi}(u_{\varepsilon_{j}},1_{B_{R_{0}}})\to 0\qquad\mbox{as
$j\to\infty$}\,,$

which gives the contradiction

$\alpha_{*}\leq\alpha_{\varepsilon_{j}}(u_{j})\leq
d_{\Phi}(u_{j},u_{\varepsilon_{j}})\leq
d_{\Phi}(u_{j},1_{B_{R_{0}}})+C\,d_{\Phi}(u_{\varepsilon_{j}},1_{B_{R_{0}}})^{(n-1)/n}\to
0^{+}\,,$

as $j\to\infty$. ∎

###  5.2. Proof of Theorem 5.1 in the radial decreasing case

We start by noticing that, thanks to the results proved in the previous
sections, we can quickly prove Theorem 5.1 for functions in $\mathcal{R}_{0}$.

######  Theorem 5.3.

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.11) and (1.12), then there
exist universal constants $C$ and $\varepsilon_{0}$ such that, for every
$\varepsilon<\varepsilon_{0}$, denoting by $u_{\varepsilon}$ the unique
minimizer of $\psi(\varepsilon)$ in $\mathcal{R}_{0}$, one has

$C\,\sqrt{\delta_{\varepsilon}(u)}\geq d_{\Phi}(u,u_{\varepsilon})\,,$ (5.5)

whenever $u\in H^{1}(\mathbb{R}^{n};[0,1])\cap\mathcal{R}_{0}$ with
$\int_{\mathbb{R}^{n}}V(u)=1$.

###### Proof.

Arguing by contradiction, we can find $\varepsilon_{j}\to 0^{+}$ and
$\\{v_{j}\\}_{j}$ in $H^{1}(\mathbb{R}^{n};[0,1])\cap\mathcal{R}_{0}$ with

$\int_{\mathbb{R}^{n}}V(v_{j})=1\,,\qquad
a_{j}=\frac{\mathcal{AC}_{\varepsilon_{j}}(v_{j})-\psi(\varepsilon_{j})}{d_{\Phi}(v_{j},u_{j})^{2}}\to
0\qquad\mbox{as $j\to\infty$}\,,$

where $u_{j}=u_{\varepsilon_{j}}$ and, thanks to Lemma 5.2 and to $a_{j}\to
0^{+}$, we have

$\lim_{j\to\infty}d_{\Phi}(v_{j},u_{j})=0\,.$ (5.6)

Correspondingly we consider the variational problems

$\gamma_{j}=\gamma(\varepsilon_{j},a_{j},v_{j})=\inf\Big{\\{}\mathcal{AC}_{\varepsilon_{j}}(w)+a_{j}\,d_{\Phi}(w,v_{j}):w\in
H^{1}(\mathbb{R}^{n};[0,1])\,,\int_{\mathbb{R}^{n}}V(w)=1\Big{\\}}\,.$

With $a_{0}$, $\ell_{0}$ and $\varepsilon_{0}$ as in Theorem 2.2, we notice
that, for $j$ large enough, we have $a_{j}\in(0,a_{0})$,
$\varepsilon_{j}<\varepsilon_{0}$, and

$\mathcal{AC}_{\varepsilon_{j}}(v_{j})\leq\psi(\varepsilon_{j})+a_{j}\,\ell_{0}\,,\qquad
d_{\Phi}(v_{j},u_{j})\leq\ell_{0}\,.$ (5.7)

In particular we can apply Theorem 2.2, and deduce the existence of minimizers
$w_{j}$ of $\gamma_{j}$. We claim that, as $j\to\infty$,

$\lim_{j\to\infty}\frac{\mathcal{AC}_{\varepsilon_{j}}(w_{j})-\psi(\varepsilon_{j})}{d_{\Phi}(w_{j},u_{j})^{2}}=0\,.$
(5.8)

To show this, we first notice that, by comparing $w_{j}$ to $u_{j}$ we have

$\mathcal{AC}_{\varepsilon_{j}}(w_{j})+a_{j}\,d_{\Phi}(w_{j},v_{j})\leq\psi(\varepsilon_{j})+a_{j}\,d_{\Phi}(u_{j},v_{j})\,,$

so that (5.6) gives $\delta_{\varepsilon_{j}}(w_{j})\to 0$, and then Lemma 5.2
implies

$\lim_{j\to\infty}d_{\Phi}(w_{j},u_{j})=0\,.$ (5.9)

Next, comparing $w_{j}$ to $v_{j}$ we find that

$\mathcal{AC}_{\varepsilon_{j}}(w_{j})+a_{j}\,d_{\Phi}(w_{j},v_{j})\leq\mathcal{AC}_{\varepsilon_{j}}(v_{j})\,,$

so that $\psi(\varepsilon_{j})\leq\mathcal{AC}_{\varepsilon_{j}}(w_{j})$ and
the definition of $a_{j}$ give

$d_{\Phi}(w_{j},v_{j})\leq\frac{\mathcal{AC}_{\varepsilon_{j}}(v_{j})-\psi(\varepsilon_{j})}{a_{j}}=d_{\Phi}(v_{j},u_{j})^{2}\,.$
(5.10)

By (2.59), (5.6), (5.9), and (5.10) we find

$\displaystyle\big{|}d_{\Phi}(w_{j},u_{j})-d_{\Phi}(v_{j},u_{j})\big{|}$
$\displaystyle\leq$ $\displaystyle
C\,\max\big{\\{}d_{\Phi}(w_{j},u_{j}),d_{\Phi}(v_{j},u_{j})\big{\\}}^{1/n}\,d_{\Phi}(w_{j},v_{j})^{(n-1)/n}$
$\displaystyle=$ $\displaystyle{\rm
o}\big{(}d_{\Phi}(v_{j},u_{j})^{2(n-1)/n}\big{)}\,,$

where $2(n-1)/n\geq 1$ thanks to $n\geq 2$. Thus, $d_{\Phi}(w_{j},u_{j})\geq
d_{\Phi}(v_{j},u_{j})/C$ for $j$ large enough, and
$\mathcal{AC}_{\varepsilon_{j}}(w_{j})\leq\mathcal{AC}_{\varepsilon_{j}}(v_{j})$
gives

$\frac{\mathcal{AC}_{\varepsilon_{j}}(w_{j})-\psi(\varepsilon_{j})}{d_{\Phi}(w_{j},u_{j})^{2}}\leq
C\,\frac{\mathcal{AC}_{\varepsilon_{j}}(v_{j})-\psi(\varepsilon_{j})}{d_{\Phi}(v_{j},u_{j})^{2}}\to
0^{+}\,,$

as claimed in (5.8).

We now derive a contradiction to (5.8). By Theorem 2.2, we know that
$w_{j}\in\mathcal{R}_{0}^{*}\cap C^{2,1/(n-1)}_{{\rm loc}}(\mathbb{R}^{n})$,
$0<w_{j}<1$ on $\mathbb{R}^{n}$, and

$-2\,\varepsilon_{j}^{2}\,\Delta
w_{j}=\varepsilon_{j}\,w_{j}\,(1-w_{j})\,\mathrm{E}_{j}-W^{\prime}(w_{j})\qquad\mbox{on
$\mathbb{R}^{n}$}\,,$ (5.11)

where $\mathrm{E}_{j}$ is a continuous radial function on $\mathbb{R}^{n}$
with

$\sup_{\mathbb{R}^{n}}|\mathrm{E}_{j}|\leq C\,.$ (5.12)

We can thus apply Theorem 3.1 to $w_{j}$. In particular, since both $u_{j}$
and $w_{j}$ obey the resolution formula (3.8), we have that
$h_{j}=w_{j}-u_{j}$ satisfies

$|h_{j}(R_{0}+\varepsilon_{j}\,s)|\leq
C\,\varepsilon_{j}\,e^{-|s|/C}\qquad\forall
s\geq-\frac{R_{0}}{\varepsilon_{j}}\,.$ (5.13)

In particular,

$\|h_{j}\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C\,\varepsilon_{j}\,,\qquad\int_{\mathbb{R}^{n}}h_{j}^{2}\leq
C\,\varepsilon_{j}\,.$

and we can thus apply Theorem 4.1 to deduce

$\displaystyle\mathcal{AC}_{\varepsilon_{j}}(w_{j})-\psi(\varepsilon_{j})$
$\displaystyle\geq$
$\displaystyle\frac{1}{C}\,\int_{\mathbb{R}^{n}}\varepsilon_{j}\,|\nabla
h_{j}|^{2}+\frac{h_{j}^{2}}{\varepsilon_{j}}$ (5.14) $\displaystyle\geq$
$\displaystyle\frac{1}{C}\,\int_{\mathbb{R}^{n}}|\nabla(h_{j}^{2})|\geq\frac{1}{C}\,\Big{(}\int_{\mathbb{R}^{n}}|h_{j}|^{2n/(n-1)}\Big{)}^{(n-1)/n}\,,$

where we have also used the $BV$-Sobolev inequality. By (5.13), and by
applying (3.14) to $u_{j}$ in combination with (A.6), we find that, if
$A_{j}=B_{R_{0}+c_{j}}\setminus B_{R_{0}-b_{j}}$, then, for every
$x\in\mathbb{R}^{n}\setminus A_{j}$ we have

$|\Phi(u_{j}(x))-\Phi(w_{j}(x))|\leq|h_{j}(x)|\,\int_{0}^{1}\,\sqrt{W(u_{j}(x)+t\,h_{j}(x))}\,dt\leq
C\,|h_{j}(x)|\,e^{-||x|-R_{0}|/C\,\varepsilon_{j}}\,,$

and, therefore,

$\displaystyle\int_{\mathbb{R}^{n}\setminus
A_{j}}|\Phi(u_{j})-\Phi(w_{j})|^{n/(n-1)}$ $\displaystyle\leq$ $\displaystyle
C\,\int_{\mathbb{R}^{n}\setminus
A_{j}}|h_{j}|^{n/(n-1)}\,e^{-||x|-R_{0}|/C\,\varepsilon_{j}}$ (5.15)
$\displaystyle\leq$
$\displaystyle\,C\,\sqrt{\varepsilon_{j}}\,\Big{(}\int_{\mathbb{R}^{n}}|h_{j}|^{2\,n/(n-1)}\Big{)}^{1/2}\,.$

If, instead, $x\in A_{j}$, then by $|\Phi(u_{j})-\Phi(w_{j})|\leq C\,|h_{j}|$
and $\mathcal{L}^{n}(A_{j})\leq C\,\varepsilon_{j}$ we find

$\int_{A_{j}}|\Phi(u_{j})-\Phi(w_{j})|^{n/(n-1)}\leq
C\,\sqrt{\varepsilon_{j}}\,\Big{(}\int_{\mathbb{R}^{n}}|h_{j}|^{2\,n/(n-1)}\Big{)}^{1/2}\,.$
(5.16)

By combining (5.14), (5.15) and (5.16), and thanks to $\varepsilon_{j}\leq 1$,
$n/(n-1)\geq 1$, and $\delta_{\varepsilon_{j}}(w_{j})\leq 1$, we conclude that

$d_{\Phi}(u_{j},w_{j})\leq
C\,\sqrt{\varepsilon_{j}}\,\delta_{\varepsilon_{j}}(w_{j})^{n/2\,(n-1)}\leq
C\,\sqrt{\delta_{\varepsilon_{j}}(w_{j})}\,,$

in contradiction to (5.8). ∎

######  Remark 5.1.

The argument we have just presented provides further indication that (5.5)
should not provide a sharp rate on radial decreasing functions. The sharp
stability estimate on small radial perturbations of $u_{\varepsilon}$ is
clearly given in Theorem 4.1, but it is not clear what form the sharp
stability estimate should take on $\mathcal{R}_{0}$ (or, more generally, on
arbitrary radial functions).

###  5.3. Reduction to radial decreasing functions

We now discuss the reduction of (5.2) to the case of radial decreasing
functions. We do this by adapting to our setting the “quantitative
symmetrization” strategy developed in [FMP08, FMP07] in the study of Euclidean
isoperimetry.

Given $n\geq 2$ and $k\in\\{1,...,n\\}$ we say that
$u:\mathbb{R}^{n}\to\mathbb{R}$ is $k$-symmetric if there exists $k$ mutually
orthogonal hyperplanes such that $u$ is symmetric by reflection through each
of these hyperplanes. The class of $n$-symmetric functions is particularly
convenient when it comes to quantify sharp inequalities involving radial
decreasing rearrangements. Consider for example the Pólya-Szegö inequality

$\int_{\mathbb{R}^{n}}|\nabla u|^{2}\geq\int_{\mathbb{R}^{n}}|\nabla
u^{*}|^{2}\,,$ (5.17)

where $u^{*}$ is the radial decreasing rearrangement of $u$. A classical
result of Brothers and Ziemer [BZ88] shows that equality can hold in (5.17)
without $u$ being a translation of $u^{*}$; in general, the additional
condition that $(u^{*})^{\prime}<0$ a.e. must be assumed to deduce symmetry
from equality in (5.17) (compare with step six in the proof of Theorem 2.1).
However, if $u$ is $n$-symmetric, then equality in (5.17) automatically
implies that $u$ is radial decreasing. A quantitative version of this
statement is proved in [FMP07, Theorem 2.2] in the $BV$-case of (5.17), and in
[CFMP09, Theorem 3] in the Sobolev case. The following theorem is an
adaptation of those results to our setting.

######  Theorem 5.4 (Reduction from $n$-symmetric to radial decreasing
functions).

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.11) and (1.12), then there
exists a universal constant $C$ with the following property. If $u\in
H^{1}(\mathbb{R}^{n};[0,1])$ is a $n$-symmetric function with
$\int_{\mathbb{R}^{n}}V(u)=1$ and $u^{*}$ is its radial decreasing
rearrangement, then

$d_{\Phi}(u,u^{*})\leq
C\,\Big{(}\int_{\mathbb{R}^{n}}W(u)\Big{)}^{1/2}\,\Big{(}\int_{\mathbb{R}^{n}}|\nabla
u|^{2}-\int_{\mathbb{R}^{n}}|\nabla u^{*}|^{2}\Big{)}^{1/2}\,.$ (5.18)

Moreover, for every $\varepsilon>0$ we have

$\alpha_{\varepsilon}(u)\leq
C\,\Big{(}\alpha_{\varepsilon}(u^{*})+\big{(}\mathcal{AC}_{\varepsilon}(u)\,\delta_{\varepsilon}(u)\big{)}^{1/2}\,\Big{)}\,.$
(5.19)

###### Proof.

We first claim that

$\displaystyle d_{\Phi}(u,u^{*})$ $\displaystyle\leq$
$\displaystyle\frac{n}{n-1}\,\int_{0}^{1}\mathcal{L}^{n}(E_{t})\Phi(t)^{1/(n-1)}\,\sqrt{W(t)}\,dt\,,$
(5.20) $\displaystyle\int_{\mathbb{R}^{n}}|\nabla
u|^{2}-\int_{\mathbb{R}^{n}}|\nabla u^{*}|^{2}$ $\displaystyle\geq$
$\displaystyle\frac{1}{C(n)}\,\int_{0}^{1}\Big{(}\frac{\mathcal{L}^{n}(E_{t})}{\mu(t)}\Big{)}^{2}\frac{\mu(t)^{2\,(n-1)/n}}{-\mu^{\prime}(t)}\,dt\,,$
(5.21)

where $E_{t}=\\{u>t\\}\Delta\\{u^{*}>t\\}$,
$\mu(t)=\mathcal{L}^{n}(\\{u>t\\})$, and $\mu^{\prime}(t)$ denotes the
absolutely continuous part of the distributional derivative of the decreasing
function $\mu$. To prove (5.20) we recall that, by [CFMP09, Lemma 5], we have

$d_{\Phi}(u,u^{*})\leq\frac{n}{n-1}\,\int_{0}^{1}\,\mathcal{L}^{n}(F_{s})\,s^{1/(n-1)}\,ds\,,$

provided $F_{s}=\\{\Phi(u)>s\\}\Delta\\{\Phi(u^{*})>s\\}$. Since $\Phi$ is
strictly increasing, we have $F_{\Phi(t)}=E_{t}$, so that the change of
variables $s=\Phi(t)$ gives (5.20). To prove (5.21) we just notice that this
is [CFMP09, Equation (3.18)]. Now, by Hölder inequality and (5.20), we find
that

$\displaystyle\int_{0}^{1}\mathcal{L}^{n}(E_{s})\,\Phi^{1/(n-1)}\,\sqrt{W}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{\mathcal{L}^{n}(E_{s})}{\mu}\,\frac{\mu^{(n-1)/n}}{(-\mu^{\prime})^{1/2}}\,\frac{(-\mu^{\prime})^{1/2}}{\mu^{-1/n}}\,\Phi^{1/(n-1)}\,\sqrt{W}$
$\displaystyle\leq$
$\displaystyle\Big{(}\int_{0}^{1}\Big{(}\frac{\mathcal{L}^{n}(E_{s})}{\mu}\Big{)}^{2}\frac{\mu^{2\,(n-1)/n}}{-\mu^{\prime}}\Big{)}^{1/2}\,\Big{(}\int_{0}^{1}\frac{-\mu^{\prime}}{\mu^{-2/n}}\,\Phi^{2/(n-1)}\,W\,\Big{)}^{1/2}\,.$

By $1=\int_{\mathbb{R}^{n}}V(u)\geq V(t)\,\mu(t)$ for every $t\in(0,1)$, we
have

$\int_{0}^{1}\frac{-\mu^{\prime}}{\mu^{-2/n}}\,\Phi^{2/(n-1)}\,W\leq\int_{0}^{1}-\mu^{\prime}\,\big{(}V\,\mu\big{)}^{2/n}\,W\leq\int_{0}^{1}\,-\mu^{\prime}\,W\leq\int_{\mathbb{R}^{n}}W(u)\,,$

where in the last inequality we have used
$-\mu^{\prime}\,d\mathcal{L}^{1}\leq-D\mu$, integration by parts and Fubini’s
theorem to deduce

$-\int_{0}^{1}\,W\,d[D\mu]=\int_{0}^{1}\,W^{\prime}(t)\,\mu(t)\,dt=\int_{\mathbb{R}^{n}}\,dx\int_{0}^{u(x)}W^{\prime}(t)\,dt=\int_{\mathbb{R}^{n}}W(u)\,.$

By combining (5.20), (5.21) and these estimates we find (5.18). To prove
(5.19), we notice that, by
$\int_{\mathbb{R}^{n}}W(u)=\int_{\mathbb{R}^{n}}W(u^{*})$ and
$\int_{\mathbb{R}^{n}}V(u^{*})=1$, (5.18) gives

$d_{\Phi}(u,u^{*})\leq
C\,\mathcal{AC}_{\varepsilon}(u)^{1/2}\,\Big{(}\mathcal{AC}_{\varepsilon}(u)-\mathcal{AC}_{\varepsilon}(u^{*})\Big{)}^{1/2}\leq
C\,\mathcal{AC}_{\varepsilon}(u)^{1/2}\,\delta_{\varepsilon}(u)^{1/2}$ (5.22)

and then (5.19) follows by the triangular inequality in
$L^{n/(n-1)}(\mathbb{R}^{n})$. ∎

Next we discuss the reduction from generic functions to $n$-symmetric ones.

######  Theorem 5.5 (Reduction to $n$-symmetric functions).

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.11) and (1.12), then there
exist universal constants $\varepsilon_{0}$ and $\delta_{0}$ with the
following property. If $u\in H^{1}(\mathbb{R}^{n};[0,1])$,
$\int_{\mathbb{R}^{n}}V(u)=1$ and $\delta_{\varepsilon}(u)\leq\delta_{0}$ for
some $\varepsilon<\varepsilon_{0}$, then there exists $v\in
H^{1}(\mathbb{R}^{n};[0,1])$ with $\int_{\mathbb{R}^{n}}V(v)=1$ such that $v$
is $n$-symmetric and

$\alpha_{\varepsilon}(u)\leq
C\,\alpha_{\varepsilon}(v)\,,\qquad\delta_{\varepsilon}(v)\leq
C\,\delta_{\varepsilon}(u)\,.$ (5.23)

###### Proof.

Without loss of generality we can assume that
$\delta_{\varepsilon}(u)\leq\delta_{0}$ for a universal constant $\delta_{0}$.
By Lemma 5.2 we can choose $\delta_{0}$ so that
$\alpha_{\varepsilon}(u)\leq\alpha_{0}$ for $\alpha_{0}$ a universal constant
of our choice. We divide the proof into a few steps.

Step one: We prove that, if $u$ is $k$-symmetric, $\\{H_{i}\\}_{i=1}^{k}$ are
the mutually orthogonal hyperplanes of symmetry of $u$, and
$J=\bigcap_{i=1}^{k}H_{i}$, then

$\alpha_{\varepsilon}(u;J)=\inf_{x\in J}\,d_{\Phi}(u,T_{x}u_{\varepsilon})\leq
C(n)\,\alpha_{\varepsilon}(u)\,.$ (5.24)

In other words, in computing the asymmetry of $u$ in the proof of an estimate
like (5.2), we can compare $u$ with a translation of $u_{\varepsilon}$ with
maximum on $J$.

Indeed, let $x_{0}\in\mathbb{R}^{n}$ be such that
$\alpha_{\varepsilon}(u)=d_{\Phi}(u,T_{x_{0}}u_{\varepsilon})$. Without loss
of generality, we can assume $x_{0}\not\in J$. In particular, if $y_{0}$
denotes the reflection of $x_{0}$ with respect to $J$, then $y_{0}\neq x_{0}$
and

$d_{\Phi}(u,T_{y_{0}}u_{\varepsilon})=d_{\Phi}(u,T_{x_{0}}u_{\varepsilon})=\alpha_{\varepsilon}(u)\,,$
(5.25)

that is, also $y_{0}$ is an optimal center for computing
$\alpha_{\varepsilon}(u)$. Let $z_{0}=(x_{0}+y_{0})/2$, so that $z_{0}\in J$,
let $\nu=(x_{0}-y_{0})/|x_{0}-y_{0}|$ (which is well defined by $x_{0}\neq
y_{0}$), and let $H$ be the open half-space orthogonal to $\nu$, containing
$x_{0}$, and such that $z_{0}\in\partial H$. By
$T_{z_{0}+t\,\nu}u_{\varepsilon}(x)=u_{\varepsilon}(x-z_{0}-t\,\nu)$, we have
that

$\frac{d}{dt}\,T_{z_{0}+t\,\nu}u_{\varepsilon}(x)=-\nu\cdot\frac{x-z_{0}-t\,\nu}{|x-z_{0}-t\,\nu|}\,u_{\varepsilon}^{\prime}(|x-x_{0}-t\,\nu|)>0\,,\qquad\forall
x\in H\,,t<0\,,$

since $u_{\varepsilon}^{\prime}<0$, and since the fact that $\nu$ points
inside $H$ gives

$\displaystyle(z-z_{0})\cdot\nu>0\,,\qquad\forall z\in H\,,$ $\displaystyle
z=x-t\,\nu\in H\,,\qquad\forall x\in H\,,t<0\,.$

We thus find that, if $t<0$,

$\displaystyle\frac{d}{dt}\,\int_{H}|\Phi(T_{x_{0}}u)-\Phi(T_{z_{0}+t\,\nu}u_{\varepsilon})|^{n/(n-1)}$
$\displaystyle=$
$\displaystyle\frac{n}{n-1}\,\int_{H}|\Phi(u)-\Phi(T_{z_{0}+t\,\nu}u_{\varepsilon})|^{1/(n-1)}\,\sqrt{W(T_{z_{0}+t\,\nu}u_{\varepsilon})}\,\frac{d}{dt}\,T_{z_{0}+t\,\nu}u_{\varepsilon}>0\,,$

so that

$\displaystyle\int_{H}|\Phi(T_{x_{0}}u)-\Phi(T_{y_{0}}u_{\varepsilon})|^{n/(n-1)}$
$\displaystyle=$
$\displaystyle\int_{H^{-}}|\Phi(T_{x_{0}}u)-\Phi(T_{z_{0}+t\,\nu}u_{\varepsilon})|^{n/(n-1)}\Big{|}_{t=-|x_{0}-y_{0}|/2}$
(5.26) $\displaystyle\leq$
$\displaystyle\int_{H}|\Phi(T_{x_{0}}u)-\Phi(T_{z_{0}+t\,\nu}u_{\varepsilon})|^{n/(n-1)}\Big{|}_{t=0}$
$\displaystyle\leq$
$\displaystyle\int_{H}|\Phi(T_{x_{0}}u)-\Phi(T_{z_{0}}u_{\varepsilon})|^{n/(n-1)}$

Now, since both $u$ and $T_{z_{0}}u_{\varepsilon}$ are symmetric by reflection
with respect to $\partial H$, we have that

$\int_{\mathbb{R}^{n}}|\Phi(u)-\Phi(T_{z_{0}}u_{\varepsilon})|^{n/(n-1)}=2\,\int_{H}|\Phi(u)-\Phi(T_{z_{0}}u_{\varepsilon})|^{n/(n-1)}\,,$
(5.27)

therefore, by (5.25), (5.26) and (5.27) we conclude that

$\displaystyle\alpha_{\varepsilon}(u;J)$ $\displaystyle\leq$ $\displaystyle
d_{\Phi}(u,T_{z_{0}}u_{\varepsilon})=2\,\int_{H}|\Phi(u)-\Phi(T_{z_{0}}u_{\varepsilon})|^{n/(n-1)}$
$\displaystyle\leq$ $\displaystyle
C(n)\,\Big{(}\int_{H}|\Phi(u)-\Phi(T_{x_{0}}u_{\varepsilon})|^{n/(n-1)}+\int_{H}|\Phi(T_{x_{0}}u_{\varepsilon})-\Phi(T_{z_{0}}u_{\varepsilon})|^{n/(n-1)}\Big{)}$
$\displaystyle\leq$ $\displaystyle
C(n)\,\Big{(}\alpha_{\varepsilon}(u)+\int_{H}|\Phi(T_{y_{0}}u_{\varepsilon})-\Phi(T_{x_{0}}u_{\varepsilon})|^{n/(n-1)}\Big{)}$
$\displaystyle\leq$ $\displaystyle
C(n)\,\Big{(}\alpha_{\varepsilon}(u)+d_{\Phi}(T_{y_{0}}u_{\varepsilon},T_{x_{0}}u_{\varepsilon})\Big{)}$
$\displaystyle\leq$ $\displaystyle
C(n)\,\Big{(}\alpha_{\varepsilon}(u)+d_{\Phi}(T_{y_{0}}u_{\varepsilon},u)+d_{\Phi}(u,T_{x_{0}}u_{\varepsilon})\Big{)}=C(n)\,\alpha_{\varepsilon}(u)\,,$

that is (5.24).

Step two: Let $H_{1}$ and $H_{2}$ be two orthogonal hyperplanes through the
origin, let $H_{i}^{\pm}$ be the half-spaces defined by $H_{i}$, and let
$x_{i}^{\pm}\in\partial H_{i}$. For $i=1,2$, consider the functions

$U[u_{\varepsilon},H_{i},x_{i}^{+},x_{i}^{-}]=1_{H_{i}^{+}}\,T_{x_{i}^{+}}u_{\varepsilon}+1_{H_{i}^{-}}\,T_{x_{i}^{-}}u_{\varepsilon}\,,$

obtained by “gluing” the restriction of $u_{\varepsilon}$ to $H_{i}^{+}$
translated by $x_{1}^{+}$ to the restriction of $u_{\varepsilon}$ to
$H_{i}^{-}$ translated by $x_{1}^{+}$ (notice that translating by
$x_{i}^{\pm}$ brings $H_{i}^{+}$ and $H_{i}^{-}$ into themselves). Setting for
brevity

$U_{\varepsilon,i}=U[u_{\varepsilon},H_{i},x_{i}^{+},x_{i}^{-}]$

we claim that, for every $a\in(0,1)$ there is $\kappa=\kappa(a,n,W)>0$ such
that if

$\max\Big{\\{}|x_{1}^{+}-x_{1}^{-}|,|x_{2}^{+}-x_{2}^{-}|,|x_{1}^{+}-x_{2}^{+}|\Big{\\}}\leq\kappa\,,$
(5.28)

then, for every $\varepsilon<\varepsilon_{0}$,

$\max\Big{\\{}d_{\Phi}(T_{x_{1}^{+}}u_{\varepsilon},T_{x_{1}^{-}}u_{\varepsilon}),d_{\Phi}(T_{x_{2}^{+}}u_{\varepsilon},T_{x_{2}^{-}}u_{\varepsilon})\Big{\\}}\leq\frac{8}{1-a}\,d_{\Phi}(U_{\varepsilon,1},U_{\varepsilon,2})\,.$
(5.29)

Indeed, since $H_{1}$ and $H_{2}$ are hyperplanes through the origin and
$u_{\varepsilon}\in\mathcal{R}_{0}$, we have

$\int_{H_{1}^{\pm}}V(T_{x_{1}^{\pm}}u_{\varepsilon})=\frac{1}{2}\,,\qquad\int_{H_{2}^{\pm}}V(T_{x_{2}^{\pm}}u_{\varepsilon})=\frac{1}{2}\,.$

It is in general not true that, say, $H_{1}^{+}\cap H_{2}^{+}$ has measure
$1/4$ for either $V(T_{x_{1}^{\pm}}u_{\varepsilon})\,dx$ or
$V(T_{x_{1}^{\pm}}u_{\varepsilon})\,dx$. However, provided we choose $\kappa$
sufficiently small, thanks to Theorem 3.1, we can definitely ensure that, for
every $\varepsilon<\varepsilon_{0}$ and $\beta,\gamma\in\\{+,-\\}$, we have

$\int_{H_{1}^{\beta}\cap
H_{2}^{\gamma}}|\Phi(T_{x_{1}^{\beta}}u_{\varepsilon})-\Phi(T_{x_{2}^{\gamma}}u_{\varepsilon})|^{n/(n-1)}\geq\frac{1-a}{4}\,d_{\Phi}(T_{x_{1}^{\beta}}u_{\varepsilon},T_{x_{2}^{\gamma}}u_{\varepsilon})\,.$

Correspondingly,

$\displaystyle d_{\Phi}(U_{\varepsilon,1},U_{\varepsilon,2})$
$\displaystyle\geq$ $\displaystyle\int_{H_{1}^{\beta}\cap
H_{2}^{\gamma}}|\Phi(U_{\varepsilon,1})-\Phi(U_{\varepsilon,2})|^{n/(n-1)}$
$\displaystyle=$ $\displaystyle\int_{H_{1}^{\beta}\cap
H_{2}^{\gamma}}|\Phi(T_{x_{1}^{\beta}}u_{\varepsilon})-\Phi(T_{x_{2}^{\gamma}}u_{\varepsilon})|^{n/(n-1)}\geq\frac{1-a}{4}\,d_{\Phi}(T_{x_{1}^{\beta}}u_{\varepsilon},T_{x_{2}^{\gamma}}u_{\varepsilon})\,,$

and thus

$\displaystyle
d_{\Phi}(T_{x_{1}^{+}}u_{\varepsilon},T_{x_{1}^{-}}u_{\varepsilon})^{(n-1)/n}$
$\displaystyle\leq$ $\displaystyle
d_{\Phi}(T_{x_{1}^{+}}u_{\varepsilon},T_{x_{2}^{+}}u_{\varepsilon})^{(n-1)/n}+d_{\Phi}(T_{x_{2}^{+}}u_{\varepsilon},T_{x_{1}^{-}}u_{\varepsilon})^{(n-1)/n}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{8}{1-a}\,d_{\Phi}(U_{\varepsilon,1},U_{\varepsilon,2})\Big{)}^{(n-1)/n}\,,$

as claimed.

Step three: Given $u\in H^{1}(\mathbb{R}^{n};[0,1])$ with
$\int_{\mathbb{R}^{n}}V(u)=1$, we now consider an hyperplane $H$ such that, if
$H^{+}$ and $H^{-}$ denote the two open half-spaces defined by $H$, then

$\int_{H^{+}}V(u)=\int_{H^{-}}V(u)=\frac{1}{2}\,.$

Denoting by $\rho_{H}$ the reflection with respect to $H$, we let

$u^{+}=1_{H^{+}}\,u+1_{H^{-}}\,(u\circ\rho_{H})\,,\qquad
u^{-}=1_{H^{-}}\,u+1_{H^{+}}\,(u\circ\rho_{H})\,,$ (5.30)

and notice that $u^{\pm}\in H^{1}(\mathbb{R}^{n};[0,1])$, with

$2\,\mathcal{AC}_{\varepsilon}(u)=\mathcal{AC}_{\varepsilon}(u^{+})+\mathcal{AC}_{\varepsilon}(u^{-})\,,\qquad\int_{\mathbb{R}^{n}}V(u^{+})=\int_{\mathbb{R}^{n}}V(u^{-})=1\,.$
(5.31)

We claim that

$\max\\{\delta_{\varepsilon}(u^{+}),\delta_{\varepsilon}(u^{-})\\}\leq
2\,\delta_{\varepsilon}(u)\,,\qquad\alpha_{\varepsilon}(u)\leq
C(n)\,\Big{\\{}\alpha_{\varepsilon}(u^{+})+\alpha_{\varepsilon}(u^{-})+d_{\Phi}(T_{x^{+}}u_{\varepsilon},T_{x^{-}}u_{\varepsilon})\Big{\\}}\,,$
(5.32)

provided $T_{x^{+}}u_{\varepsilon}=$ and
$T_{x^{-}}u_{\varepsilon}=T_{x^{-}}u_{\varepsilon}$ are such that
$x^{+},x^{-}\in H$ with

$\alpha_{\varepsilon}(u^{+};H)=d_{\Phi}(u^{+},T_{x^{+}}u_{\varepsilon})\,,\qquad\alpha_{\varepsilon}(u^{-};H)=d_{\Phi}(u^{-},T_{x^{-}}u_{\varepsilon})\,.$

The first inequality in (5.32) is obvious from (5.31). To prove the second one
we notice that

$\displaystyle\alpha_{\varepsilon}(u)$ $\displaystyle\leq$ $\displaystyle
d_{\Phi}(u,T_{x^{+}}u_{\varepsilon})=\int_{H^{+}}|\Phi(u)-\Phi(T_{x^{+}}u_{\varepsilon})|^{n/(n-1)}+\int_{H^{-}}|\Phi(u)-\Phi(T_{x^{+}}u_{\varepsilon})|^{n/(n-1)}$
$\displaystyle=$
$\displaystyle\int_{H^{+}}|\Phi(u^{+})-\Phi(T_{x^{+}}u_{\varepsilon})|^{n/(n-1)}+\int_{H^{-}}|\Phi(u^{-})-\Phi(T_{x^{+}}u_{\varepsilon})|^{n/(n-1)}$
$\displaystyle\leq$ $\displaystyle
C(n)\,\Big{\\{}d_{\Phi}(u^{+},T_{x^{+}}u_{\varepsilon})+d_{\Phi}(u^{-},T_{x^{-}}u_{\varepsilon})+d_{\Phi}(T_{x^{-}}u_{\varepsilon},T_{x^{+}}u_{\varepsilon})\Big{\\}}\,,$

that is the second inequality in (5.32).

With these preliminary considerations in place, we now prove that if $u\in
H^{1}(\mathbb{R}^{n};[0,1])$ with $\int_{\mathbb{R}^{n}}V(u)=1$, if $H_{1}$
and $H_{2}$ are orthogonal hyperplanes such that the corresponding half-spaces
$H_{i}^{\pm}$ satisfy

$\int_{H_{i}^{\pm}}V(u)=\frac{1}{2}\,,$

if $u_{i}^{\pm}$ as in (5.30) starting from $H_{i}$, then there is at least
one $v\in\\{u_{1}^{+},u_{1}^{-},u_{2}^{+},u_{2}^{-}\\}$ such that (5.23)
holds. Given that $\delta_{\varepsilon}(v)\leq 2\,\delta_{\varepsilon}(u)$ for
every $v\in\\{u_{1}^{+},u_{1}^{-},u_{2}^{+},u_{2}^{-}\\}$, we need to show
that

$\mbox{$\exists v\in\\{u_{1}^{+},u_{1}^{-},u_{2}^{+},u_{2}^{-}\\}$ such that
$\alpha_{\varepsilon}(u)\leq C\,\alpha_{\varepsilon}(v)$}\,.$ (5.33)

Denoting by $x_{i}^{\pm}$ the points in $H_{i}$ such that

$\alpha_{\varepsilon}(u_{i}^{\pm};H_{i})=d_{\Phi}(u_{i}^{\pm},T_{x_{i}^{\pm}}u_{\varepsilon})\,,$

we notice that (5.33) follows if we can show that, provided $\alpha_{0}$ is
small enough, then

either $\displaystyle\qquad
d_{\Phi}(T_{x_{1}^{+}}u_{\varepsilon},T_{x_{1}^{-}}u_{\varepsilon})\leq
M\,\Big{\\{}\alpha_{\varepsilon}(u_{1}^{+};H_{1})+\alpha_{\varepsilon}(u_{1}^{-};H_{1})\Big{\\}}$
(5.34) or $\displaystyle\qquad
d_{\Phi}(T_{x_{2}^{+}}u_{\varepsilon},T_{x_{2}^{-}}u_{\varepsilon})\leq
M\,\Big{\\{}\alpha_{\varepsilon}(u_{2}^{+};H_{2})+\alpha_{\varepsilon}(u_{2}^{-};H_{2})\Big{\\}}\,,$
(5.35)

for a constant $M$ (as it turns out, any $M>16$ works). Indeed, if, for
example, (5.34) holds, then (5.24) and (5.32) with $H=H_{1}$ give

$\alpha_{\varepsilon}(u)\leq
C\,\Big{\\{}\alpha_{\varepsilon}(u_{1}^{+})+\alpha_{\varepsilon}(u_{1}^{-})+\alpha_{\varepsilon}(u_{1}^{+};H_{1})+\alpha_{\varepsilon}(u_{1}^{-};H_{1})\Big{\\}}\leq
C\,\Big{\\{}\alpha_{\varepsilon}(u_{1}^{+})+\alpha_{\varepsilon}(u_{1}^{-})\Big{\\}}\,,$

and then either
$C\,\alpha_{\varepsilon}(u_{1}^{+})\geq\alpha_{\varepsilon}(u)$ or
$C\,\alpha_{\varepsilon}(u_{2}^{+})\geq\alpha_{\varepsilon}(u)$; in
particular, (5.33) holds. We now want to prove that either (5.34) or (5.35)
holds. We argue by contradiction. Recalling that
$\alpha_{\varepsilon}(u_{i}^{\pm};H_{i})=d_{\Phi}(u_{i}^{\pm},T_{x_{i}^{\pm}}u_{\varepsilon})$,
let us thus assume that both

$\displaystyle
d_{\Phi}(T_{x_{1}^{+}}u_{\varepsilon},T_{x_{1}^{-}}u_{\varepsilon})>M\,\Big{\\{}d_{\Phi}(u_{1}^{+},T_{x_{1}^{+}}u_{\varepsilon})+d_{\Phi}(u_{1}^{-},T_{x_{1}^{-}}u_{\varepsilon})\Big{\\}}\,,$
(5.36) $\displaystyle
d_{\Phi}(T_{x_{2}^{+}}u_{\varepsilon},T_{x_{2}^{-}}u_{\varepsilon})>M\,\Big{\\{}d_{\Phi}(u_{2}^{+},T_{x_{2}^{+}}u_{\varepsilon})+d_{\Phi}(u_{2}^{-},T_{x_{2}^{-}}u_{\varepsilon})\Big{\\}}\,,$
(5.37)

hold for $M$ to be determined. In particular, if $U_{\varepsilon,i}$, $i=1,2$,
are defined as in step two, and $\alpha_{0}$ is small enough that (5.28)
holds, then, by (5.29), we have

$\displaystyle\max\Big{\\{}d_{\Phi}(T_{x_{1}^{+}}u_{\varepsilon},T_{x_{1}^{-}}u_{\varepsilon}),d_{\Phi}(T_{x_{2}^{+}}u_{\varepsilon},T_{x_{2}^{-}}u_{\varepsilon})\Big{\\}}^{(n-1)/n}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{8}{1-a}\,d_{\Phi}(U_{\varepsilon,1},U_{\varepsilon,2})\Big{)}^{(n-1)/n}\leq\Big{(}\frac{8}{1-a}\Big{)}^{(n-1)/n}\,\sum_{i=1}^{2}d_{\Phi}(U_{\varepsilon,i},u)^{(n-1)/n}$
$\displaystyle=$
$\displaystyle\Big{(}\frac{8}{1-a}\Big{)}^{(n-1)/n}\,\sum_{i=1}^{2}\,\Big{(}\sum_{\beta=+,-}\int_{H_{i}^{\beta}}|\Phi(T_{x_{i}^{\beta}}u_{\varepsilon})-\Phi(u_{i}^{\beta})|^{n/(n-1)}\Big{)}^{(n-1)/n}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{8}{M\,(1-a)}\Big{)}^{(n-1)/n}\,\sum_{i=1}^{2}\,\Big{(}d_{\Phi}(T_{x_{i}^{+}}u_{\varepsilon},T_{x_{i}^{-}}u_{\varepsilon})\Big{)}^{(n-1)/n}$
$\displaystyle\leq$
$\displaystyle\Big{(}\frac{16}{M\,(1-a)}\Big{)}^{(n-1)/n}\,\max\Big{\\{}d_{\Phi}(T_{x_{1}^{+}}u_{\varepsilon},T_{x_{1}^{-}}u_{\varepsilon}),d_{\Phi}(T_{x_{2}^{+}}u_{\varepsilon},T_{x_{2}^{-}}u_{\varepsilon})\Big{\\}}^{(n-1)/n}\,.$

We fix $M>16$ and apply the above with $a\in(0,1)$ such that $M\,(1-a)>16$. We
find that either $x_{1}^{+}=x_{1}^{-}$ (a contradiction to (5.36)), or
$x_{2}^{+}=x_{2}^{-}$ (a contradiction to (5.37)).

Step four: We now pick a family of $n$ mutually orthogonal hyperplanes
$\\{H_{i}\\}_{i=1}^{n}$ such that, denoting by $H_{i}^{\pm}$ the corresponding
half-spaces, we have

$\int_{H_{i}^{\pm}}V(u)=\frac{1}{2}\qquad\forall i=1,...,n\,.$

Considering the hyperplanes in pairs and arguing inductively on step three, up
to a relabeling we reduce to a situation where there exists a function $v$,
symmetric by reflection with respect to each $H_{i}$, $i=1,...,n-1$, and such
that

$\alpha_{\varepsilon}(u)\leq
C\,\alpha_{\varepsilon}(v)\,,\qquad\delta_{\varepsilon}(v)\leq
2^{n}\,\delta_{\varepsilon}(v)\,,\qquad\int_{H_{n}^{\pm}}V(v)=\frac{1}{2}\,.$

We can thus consider the functions $v^{\pm}$ obtained by reflecting $v$ with
respect to $H_{n}$ as in step three. By (5.32) we have

$\max\\{\delta_{\varepsilon}(v^{+}),\delta_{\varepsilon}(v^{-})\\}\leq
2\,\delta_{\varepsilon}(v)\,,\qquad\alpha_{\varepsilon}(u)\leq
C(n)\,\Big{\\{}\alpha_{\varepsilon}(v^{+})+\alpha_{\varepsilon}(v^{-})+d_{\Phi}(T_{x^{+}}u_{\varepsilon},T_{x^{-}}u_{\varepsilon})\Big{\\}}\,,$

where $x^{+}$ and $x^{-}$ are optimal centers for
$\alpha_{\varepsilon}(v^{+};\bigcap_{i=1}^{n}H_{i})$ and
$\alpha_{\varepsilon}(v^{-};\bigcap_{i=1}^{n}H_{i})$. However,
$\bigcap_{i=1}^{n}H_{i}$ is a point, therefore $x^{+}=x^{-}$ and we have
actually proved

$\alpha_{\varepsilon}(u)\leq
C(n)\,\Big{\\{}\alpha_{\varepsilon}(v^{+})+\alpha_{\varepsilon}(v^{-})\Big{\\}}\,.$

Either $v^{+}$ or $v^{-}$ is an $n$-symmetric function with the required
properties. ∎

###  5.4. Proof of Theorem 5.1

We finally prove Theorem 5.1. By Theorem 5.5 we can directly assume that $u$
is $n$-symmetric. Hence, by Theorem 5.4, we can directly assume that
$u\in\mathcal{R}_{0}$. For $u\in\mathcal{R}_{0}$, the conclusion follows from
Theorem 5.3. Theorem 5.1 is proved.

##  6\. Proof of the Alexandrov-type theorem

In this section we complete the proof of Theorem 1.1, including in particular
proof of the Alexandrov-type result of part (iv) of the statement. We begin by
proving some of the properties of $\Psi(\sigma,m)$ stated in Theorem 1.1-(i)
and not yet discussed. We then review, in section 6.2, some classical
uniqueness and symmetry results for semilinear PDEs in relation to our
setting. Finally, in section 6.3 we review how the various results of the
paper combines into Theorem 1.1.

###  6.1. Some properties of $\Psi(\sigma,m)$

We prove here the properties of $\Psi(\sigma,m)$ stated in Theorem 1.1–(ii).
As explained in the introduction, these properties will be crucial in proving
Theorem 1.1-(iv).

######  Theorem 6.1.

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.11) and (1.12), then there
exists a universal constant $\varepsilon_{0}$ such that, setting

$\mathcal{X}(\varepsilon_{0})=\big{\\{}(\sigma,m):0<\sigma<\varepsilon_{0}\,m^{1/n}\big{\\}}\,,$

the following holds:

(i): for every $\sigma>0$, $\Psi(\sigma,\cdot)$ is concave on $(0,\infty)$; it
is strictly concave on $(0,\infty)$ in $n\geq 3$ and on
$((\sigma/\varepsilon_{0})^{n},\infty)$ if $n=2$;

(ii): $\Lambda(\sigma,m)$ is continuous on $\mathcal{X}(\varepsilon_{0})$ and

$\Big{|}m^{1/n}\,\Lambda(\sigma,m)-2\,(n-1)\,\omega_{n}^{1/n}\Big{|}\leq
C\,\frac{\sigma}{m^{1/n}}\,,\qquad\forall(\sigma,m)\in\mathcal{X}(\varepsilon_{0})\,.$
(6.1)

(iii): $\Psi(\sigma,\cdot)$ is differentiable with

$\frac{\partial\Psi}{\partial
m}(\sigma,m)=\Lambda(\sigma,m)\qquad\forall(\sigma,m)\in\mathcal{X}(\varepsilon_{0})\,.$
(6.2)

In particular, for every $\sigma>0$

$\displaystyle\mbox{$\Psi(\sigma,\cdot)$ is strictly increasing on
$((\sigma/\varepsilon_{0})^{n},\infty)$}\,,$
$\displaystyle\mbox{$\Lambda(\sigma,\cdot)$ is strictly decreasing
$((\sigma/\varepsilon_{0})^{n},\infty)$}\,.$

(iv): for every $m>0$, $\Psi(\cdot,m)$ is increasing on
$(0,\varepsilon_{0}\,m^{1/n})$.

###### Proof.

We recall for convenience the scaling formulas

$\displaystyle\int_{\mathbb{R}^{n}}f(\rho_{t}u)$ $\displaystyle=$
$\displaystyle\frac{1}{t}\,\int_{\mathbb{R}^{n}}f(u)\,,$ (6.3)
$\displaystyle\int_{\mathbb{R}^{n}}|\nabla(\rho_{t}u)|^{2}$ $\displaystyle=$
$\displaystyle t^{(2/n)-1}\,\int_{\mathbb{R}^{n}}|\nabla u|^{2}\,,$
$\displaystyle\mathcal{AC}_{\varepsilon}(\rho_{t}u)$ $\displaystyle=$
$\displaystyle\varepsilon\,t^{(2/n)-1}\,\int_{\mathbb{R}^{n}}|\nabla
u|^{2}+\frac{1}{\varepsilon\,t}\int_{\mathbb{R}^{n}}W(u)=\frac{\mathcal{AC}_{\varepsilon\,t^{1/n}}(u)}{t^{(n-1)/n}}\,.$
(6.4) $\displaystyle\Psi(\sigma,m)$ $\displaystyle=$ $\displaystyle
m^{(n-1)/n}\,\psi\Big{(}\frac{\sigma}{m^{1/n}}\Big{)}\,,$

where $\rho_{t}u(x)=u(t^{1/n}\,x)$ for $x\in\mathbb{R}^{n}$ and $t>0$, and the
divide the argument in a few steps.

Step one: We prove the concavity of $\Psi(\sigma,\cdot)$. Given
$m_{2}>m_{1}>0$, $t\in(0,1)$, $\sigma>0$, and a minimizing sequence
$\\{w_{j}\\}_{j}$ for $\Psi(\sigma,t\,m_{1}+(1-t)\,m_{2})$, we set

$\alpha_{1}=\frac{t\,m_{1}+(1-t)\,m_{2}}{m_{1}}\,,\qquad\alpha_{2}=\frac{t\,m_{1}+(1-t)\,m_{2}}{m_{2}}\,,$

so that $t/\alpha_{1}+(1-t)/\alpha_{2}=1$. Since $\rho_{\alpha_{1}}w_{j}$ and
$\rho_{\alpha_{2}}w_{j}$ are competitors for $\Psi(\sigma,m_{1})$ and
$\Psi(\sigma,m_{2})$ respectively, by the concavity of $t\mapsto t^{(n-2)/n}$
(strict if $n\geq 3$), we see that

$\displaystyle t\,\Psi(\sigma,m_{1})+(1-t)\,\Psi(\sigma,m_{2})\leq
t\,\mathcal{AC}_{\sigma}(\rho_{\alpha_{1}}w_{j})+(1-t)\,\mathcal{AC}_{\sigma}(\rho_{\alpha_{2}}w_{j})$
(6.5)
$\displaystyle=\frac{t}{\alpha_{1}}\,\Big{(}\int_{\mathbb{R}^{n}}\sigma\,\alpha_{1}^{2/n}\,|\nabla
w_{j}|^{2}+\frac{W(w_{j})}{\sigma}\Big{)}+\frac{1-t}{\alpha_{2}}\,\Big{(}\int_{\mathbb{R}^{n}}\sigma\,\alpha_{2}^{2/n}\,|\nabla
w_{j}|^{2}+\frac{W(w_{j})}{\sigma}\Big{)}$
$\displaystyle=\mathcal{AC}_{\sigma}(w_{j})+\Big{(}t\,\Big{(}\frac{1}{\alpha_{1}}\Big{)}^{(n-2)/n}+(1-t)\,\Big{(}\frac{1}{\alpha_{2}}\Big{)}^{(n-2)/n}-1\Big{)}\,\sigma\int_{\mathbb{R}^{n}}|\nabla
w_{j}|^{2}$ (6.6) $\displaystyle\leq\mathcal{AC}_{\sigma}(w_{j})\,.$ (6.7)

Letting $j\to\infty$ we deduce the concavity of $\Psi(\sigma,\cdot)$ on
$(0,\infty)$ (strict, if $n\geq 3$). If $n=2$ and
$m_{1}\geq(\sigma/\varepsilon_{0})^{n}$, then by Theorem 2.1 we can replace
the minimizing sequence $\\{w_{j}\\}_{j}$ in the above argument with a
minimizer $w$ of $\Psi(\sigma,t\,m_{1}+(1-t)\,m_{2})$. Since $w$ solves the
Euler-Lagrange equation (1.9), there cannot be a $t\neq 1$ such that
$\rho_{t}w$ solves (1.9) with the same $\sigma$ and some $t\in\mathbb{R}$.
Thus, $\rho_{\alpha_{i}}w$ cannot be a minimizer of $\Psi(\sigma,m_{i})$, and
therefore we have a strict inequality in (6.5), and no need to take a limit in
(6.7) (since $\mathcal{AC}_{\sigma}(w)=\Psi(\sigma,t\,m_{1}+(1-t)\,m_{2})$).

Step two: By Theorem 2.1 and Corollary 4.2 for every $m>0$ and
$\sigma<\varepsilon_{0}\,m^{1/n}$ there exists a unique
$u_{\sigma,m}\in\mathcal{R}_{0}$ such that $u_{\sigma,m}$ is a minimizer of
$\Psi(\sigma,m)$ and every other minimizer of $\Psi(\sigma,m)$ is a
translation of $u_{\sigma,m}$. Moreover, for some $\Lambda(\sigma,m)>0$ such
that

$-2\,\sigma^{2}\,\Delta
u_{\sigma,m}=\sigma\,\Lambda(\sigma,m)\,V^{\prime}(u_{\sigma,m})-W^{\prime}(u_{\sigma,m})\,,\qquad\mbox{on
$\mathbb{R}^{n}$}\,.$

If $u_{\varepsilon}$ denotes as usual the unique minimizer of
$\psi(\varepsilon)$ in $\mathcal{R}_{0}$, then by (6.3) and (6.4) we find

$u_{\sigma,m}=\rho_{1/m}\,u_{\varepsilon}\,,\qquad\varepsilon=\frac{\sigma}{m^{1/n}}\,,$

and thus

$\Lambda(\sigma,m)=\frac{\lambda(\varepsilon)}{m^{1/n}}\,,\qquad\varepsilon=\frac{\sigma}{m^{1/n}}\,.$
(6.8)

By combining (6.8) with Corollary 4.2 and with (4.7) we thus find that
$\Lambda$ is continuous on $\mathcal{X}(\varepsilon_{0})$, with

$\Big{|}\Lambda(\sigma,m)-\frac{2\,(n-1)\,\omega_{n}^{1/n}}{m^{1/n}}\Big{|}\leq
C\,\frac{\sigma}{m^{2/n}}\,.$ (6.9)

Step three: We prove statement (iii). For
$(\sigma,m)\in\mathcal{X}(\varepsilon_{0})$, is we set

$a(t)=\mathcal{AC}_{\sigma}((1+t)\,u_{\sigma,m})\,,\qquad
m(t)=\int_{\mathbb{R}^{n}}V((1+t)\,u_{\sigma,m})$

then

$m^{\prime}(0)=\int_{\mathbb{R}^{n}}\Phi(u_{\sigma,m})^{1/(n-1)}\,\sqrt{W(u_{\sigma,m})}\,u_{\sigma,m}>0$

and thus there exist $t_{*}>0$ and an open interval $I$ of $m$ such that $m$
is strictly increasing from $(-t_{*},t_{*})$ to $I$ with $m(0)=m$. From
$\Psi(\sigma,m(t))\leq a(t)$ for every $|t|<t_{*}$ and from that fact that $a$
is differentiable on $(-t_{*},t_{*})$ we deduce that, if $m$ is such that
$\Psi(\sigma,\cdot)$ is differentiable at $m$, then

$\frac{\partial\Psi}{\partial
m}(\sigma,m)=\frac{a^{\prime}(0)}{m^{\prime}(0)}=\frac{\int_{\mathbb{R}^{n}}2\,\nabla
u_{\sigma,m}\cdot\nabla
u_{\sigma,m}+W^{\prime}(u_{\sigma,m})\,u_{\sigma,m}}{\int_{\mathbb{R}^{n}}V^{\prime}(u_{\sigma,m})\,u_{\sigma,m}}=\Lambda(\sigma,m)\,.$

Now, by statement (i), $\Psi(\sigma,\cdot)$ is differentiable a.e. on
$((\sigma/\varepsilon_{0})^{n},\infty)$, as well as absolutely continuous,
while $\Lambda(\sigma,\cdot)$ is continuous on
$((\sigma/\varepsilon_{0})^{n},\infty)$: by the fundamental theorem of
calculus we thus conclude that $(\partial\Psi/\partial m)(\sigma,\cdot)$
exists for every $m>(\sigma/\varepsilon_{0})^{n}$ and agrees with
$\Lambda(\sigma,m)$.

Step four: We prove statement (iv). Recalling that

$\Psi(\sigma,m)=m^{(n-1)/n}\,\psi\Big{(}\frac{\sigma}{m^{1/n}}\Big{)}\,,\qquad\forall\sigma,m>0\,,$
(6.10)

we see that, since $\Psi(\sigma,\cdot)$ is differentiable on
$((\sigma/\varepsilon_{0})^{n},\infty)$, then $\psi$ is differentiable on
$(0,\varepsilon_{0})$. Since $\psi$ is differentiable on $(0,\varepsilon_{0}$,
by (6.10) we see that $\Psi(\cdot,m)$ is differentiable on
$(0,\varepsilon_{0}\,m^{1/n})$ for every $m>0$, with

$\frac{\partial\Psi}{\partial\sigma}=m^{(n-2)/n}\,\psi^{\prime}\Big{(}\frac{\sigma}{m^{1/n}}\Big{)}\,.$

Statement (iv) will thus follow by proving that $\psi^{\prime}>0$ on
$(0,\varepsilon_{0})$. To derive a useful formula for $\psi$ we differentiate
(6.10) in $m$ and use (6.2) and
$\lambda(\sigma/m^{1/n})=m^{1/n}\Lambda(\sigma,m)$ to find that

$\displaystyle\frac{n-1}{n}\,\frac{1}{m^{1/n}}\,\psi\Big{(}\frac{\sigma}{m^{1/n}}\Big{)}-\frac{1}{n}\,\frac{\sigma}{m^{2/n}}\,\psi^{\prime}\Big{(}\frac{\sigma}{m^{1/n}}\Big{)}=\frac{\lambda(\sigma/m^{1/n})}{m^{1/n}}\,.$

In particular, by (4.5),

$\displaystyle\varepsilon\,\psi^{\prime}(\varepsilon)=(n-1)\,\psi(\varepsilon)-n\,\lambda(\varepsilon)=\varepsilon\,\int_{\mathbb{R}^{n}}|\nabla
u_{\varepsilon}|^{2}-\frac{1}{\varepsilon}\int_{\mathbb{R}^{n}}W(u_{\varepsilon})\,.$

By (3.8), if we set $\eta_{\varepsilon}(s)=\eta(s-\tau_{\varepsilon})$ and
change variables according to $|x|=R_{0}+\varepsilon\,s$ we find

$\varepsilon\,\psi^{\prime}(\varepsilon)=\int_{-R_{0}/\varepsilon}^{\infty}\,\Big{\\{}\Big{(}\eta_{\varepsilon}^{\prime}+f_{\varepsilon}^{\prime}\Big{)}^{2}-W\big{(}\eta_{\varepsilon}+f_{\varepsilon}\big{)}\Big{\\}}\,(R_{0}+\varepsilon\,s)^{n-1}\,ds\,.$
(6.11)

Multiplying by $u_{\varepsilon}^{\prime}$ and then integrating on $(r,\infty)$
the Euler-Lagrange equation

$-2\,\varepsilon^{2}\,\Big{\\{}u_{\varepsilon}^{\prime\prime}+(n-1)\,\frac{u_{\varepsilon}^{\prime}}{r}\Big{\\}}=\varepsilon\,\lambda(\varepsilon)\,V^{\prime}(u_{\varepsilon})-W^{\prime}(u_{\varepsilon})\,,$

we obtain as usual

$\varepsilon^{2}\,(u_{\varepsilon}^{\prime})^{2}-2\,(n-1)\,\varepsilon^{2}\,\int_{r}^{\infty}\,\frac{(u_{\varepsilon}^{\prime})^{2}}{\rho}\,d\rho=W(u_{\varepsilon})-\varepsilon\,\lambda(\varepsilon)\,V(u_{\varepsilon})\,,$

for every $r>0$; by the change of variables $r=R_{0}+\varepsilon\,s$ we thus
find

$(\eta_{\varepsilon}^{\prime}+f_{\varepsilon}^{\prime})^{2}-2\,(n-1)\,\varepsilon\,\int_{s}^{\infty}\,\frac{(\eta_{\varepsilon}^{\prime}+f_{\varepsilon}^{\prime})^{2}}{R_{0}+\varepsilon\,t}\,dt=W(\eta_{\varepsilon}+f_{\varepsilon})-\lambda(\varepsilon)\,\varepsilon\,V(\eta_{\varepsilon}+f_{\varepsilon})$

for every $s\in(-R_{0}/\varepsilon,\infty)$. We combine this identity into
(6.11) to find

$\varepsilon\,\psi^{\prime}(\varepsilon)=\int_{-R_{0}/\varepsilon}^{\infty}\,\Big{\\{}2\,(n-1)\,\varepsilon\,\int_{s}^{\infty}\,\frac{(\eta_{\varepsilon}^{\prime}+f_{\varepsilon}^{\prime})^{2}}{R_{0}+\varepsilon\,t}\,dt-\lambda(\varepsilon)\,\varepsilon\,V(\eta_{\varepsilon}+f_{\varepsilon})\Big{\\}}\,(R_{0}+\varepsilon\,s)^{n-1}\,ds\,$
(6.12)

We now notice that, by (A.16), (A.18), and (3.9) (that is, by the exponential
decay of $\eta$, $\eta^{\prime}$, $\eta^{\prime\prime}$ and by
$|f_{\varepsilon}(s)|\leq C\,\varepsilon\,e^{-|s|/C\,\varepsilon}$ for
$s>-R_{0}/\varepsilon$), we have

$\displaystyle\int_{s}^{\infty}\,\frac{(\eta_{\varepsilon}^{\prime}+f_{\varepsilon}^{\prime})^{2}-(\eta_{\varepsilon}^{\prime})^{2}}{R_{0}+\varepsilon\,t}\,dt$
$\displaystyle\geq$ $\displaystyle
2\,\int_{s}^{\infty}\,\frac{\eta_{\varepsilon}^{\prime}\,f_{\varepsilon}^{\prime}}{R_{0}+\varepsilon\,t}\,dt$
$\displaystyle=$
$\displaystyle-2\,\frac{\eta_{\varepsilon}^{\prime}(s)\,f_{\varepsilon}(s)}{R_{0}+\varepsilon\,s}-2\,\int_{s}^{\infty}\,f_{\varepsilon}(s)\,\Big{(}\frac{\eta_{\varepsilon}^{\prime}}{R_{0}+\varepsilon\,t}\Big{)}^{\prime}\,dt$
$\displaystyle\geq$ $\displaystyle-C\,\varepsilon\,e^{-|s|/C}$

so that (6.12) gives

$\varepsilon\,\psi^{\prime}(\varepsilon)\geq\int_{-R_{0}/\varepsilon}^{\infty}\,\Big{\\{}2\,(n-1)\,\varepsilon\,\int_{s}^{\infty}\,\frac{(\eta_{\varepsilon}^{\prime})^{2}\,dt}{R_{0}+\varepsilon\,t}-\lambda(\varepsilon)\,\varepsilon\,V(\eta_{\varepsilon}+f_{\varepsilon})\Big{\\}}\,(R_{0}+\varepsilon\,s)^{n-1}\,ds-C\,\varepsilon^{2}\,.$
(6.13)

By (4.7), (3.9), $R_{0}=\omega_{n}^{-1/n}$ and (6.13), we have

$\psi^{\prime}(\varepsilon)\geq
2\,(n-1)\,\omega_{n}^{1/n}\,\int_{-R_{0}/\varepsilon}^{\infty}\,\Big{\\{}\int_{s}^{\infty}\,(\eta_{\varepsilon}^{\prime})^{2}\,dt-V(\eta_{\varepsilon})\Big{\\}}\,(R_{0}+\varepsilon\,s)^{n-1}\,ds-C\,\varepsilon\,.$
(6.14)

Since
$\int_{s}^{\infty}(\eta_{\varepsilon}^{\prime})^{2}=\Phi(\eta_{\varepsilon}(s))$
thanks to
$\eta_{\varepsilon}^{\prime}=-\sqrt{W(\eta_{\varepsilon})}=-\Phi^{\prime}(\eta_{\varepsilon})$,
by (6.14) we have

$\displaystyle\psi^{\prime}(\varepsilon)$ $\displaystyle\geq$ $\displaystyle
2\,(n-1)\,\omega_{n}^{1/n}\,\int_{\mathbb{R}}\big{(}\Phi(\eta_{\varepsilon})-V(\eta_{\varepsilon})\big{)}\,(R_{0}+\varepsilon\,s)^{n-1}\,ds-C\,\varepsilon$
$\displaystyle\geq$ $\displaystyle
2\,(n-1)\,\omega_{n}^{1/n}\,R_{0}^{n-1}\,\int_{\mathbb{R}}\big{(}\Phi(\eta)-V(\eta)\big{)}\,ds-C\,\varepsilon\,.$

Since $\Phi$ takes values in $(0,1)$, $V=\Phi^{n/(n-1)}<\Phi$ on $(0,1)$, and

$\int_{\mathbb{R}}\big{(}\Phi(\eta)-V(\eta)\big{)}\,ds$

is a universal constant. In particular, $\psi^{\prime}(\varepsilon)\geq 1/C$
for every $\varepsilon<\varepsilon_{0}$. ∎

###  6.2. General criteria for radial symmetry and uniqueness

In this brief section we exploit two classical results from [GNN81] and [PS83]
to deduce a symmetry and uniqueness result for the kind of semilinear PDE
arising as the Euler-Lagrange equation of $\Psi(\sigma,m)$.

######  Theorem 6.2.

Let $n\geq 2$, let $W\in C^{2,1}[0,1]$ satisfy (1.11) and (1.12), and consider
$\ell\in\mathbb{R}$ and $\sigma>0$.

(i): if $u\in C^{2}(\mathbb{R}^{n};[0,1])$ is a non-zero solution to

$-2\,\sigma^{2}\,\Delta
u=\sigma\,\ell\,V^{\prime}(u)-W^{\prime}(u)\qquad\mbox{on
$\mathbb{R}^{n}$}\,,$ (6.15)

with $u(x)\to 0$ as $|x|\to\infty$, then $0<u<1$ on $\mathbb{R}^{n}$ and
$u\in\mathcal{R}_{0}^{*}$.

(ii): there exists a universal constant $\nu_{0}$ such that, if
$0<\sigma\,\ell<\nu_{0}$, then, modulo translation, (6.15) has a unique
solution among functions $u\in\mathcal{R}_{0}^{*}$, with $u(x)\to 0$ as
$|x|\to\infty$ and $0<u<1$ on $\mathbb{R}^{n}$.

######  Remark 6.1.

Notice that the smallness of $\sigma\,\ell$ is required only for proving
statement (ii).

###### Proof.

Step one: We prove statement (i). We intend to apply the following particular
case of [GNN81, Theorem 2]: if $n\geq 2$, $u\in C^{2}(\mathbb{R}^{n};[0,1])$,
$u>0$ on $\mathbb{R}^{n}$, $u(x)\to 0$ as $|x|\to\infty$, $-\Delta
u+m\,u=g(u)$ on $\mathbb{R}^{n}$ with $m>0$ and $g\in C^{1}[0,1]$ with
$g(t)={\rm O}(t^{1+\alpha})$ as $t\to 0^{+}$ for some $\alpha>0$, then, up to
translations, $u\in\mathcal{R}_{0}^{*}$.

To this end we reformulate (6.15) as

$-\Delta u+m\,u=g(u)\qquad\mbox{on $\mathbb{R}^{n}$}\,,$ (6.16)

where $m=W^{\prime\prime}(0)/2\,\sigma^{2}>0$ and

$g(t)=\frac{\ell\,V^{\prime}(t)}{2\,\sigma}+\frac{W^{\prime\prime}(0)\,t-W^{\prime}(t)}{2\,\sigma^{2}}\,,\qquad
t\in[0,1]\,.$

As noticed in section A.3, $V\in C^{2,\gamma}[0,1]$ for some $\gamma\in(0,1]$,
while $W\in C^{2,1}[0,1]$: in particular $g\in C^{1}[0,1]$. By $W\in
C^{2,1}[0,1]$ with $W^{\prime}(0)=0$ we have
$|W^{\prime}(t)-W^{\prime\prime}(0)\,t||\leq C\,t^{2}$, while (A.11) states
that $|V^{\prime}(t)|\leq C\,t^{1+\alpha}$ for $t\in[0,1]$ for some
$\alpha>0$, so that

$|g(t)|\leq C(n,W,\ell,\sigma)\,t^{1+\alpha}\,,\qquad\forall t\in[0,1]\,.$
(6.17)

To check that $u>0$ on $\mathbb{R}^{n}$, we notice that, by (6.17), for every
$m^{\prime}\in(0,m)$, we can find $t_{0}>0$ such that (6.16) implies that
$-\Delta u+m^{\prime}\,u\geq 0$ on the open set $\\{u<t_{0}\\}$. Since $u\geq
0$ and $u$ is non-zero, we conclude by the strong maximum principle that $u>0$
on $\\{u<t_{0}\\}$, and thus, on $\mathbb{R}^{n}$. We are thus in the position
to apply the stated particular case of [GNN81, Theorem 2] and conclude that
$u\in\mathcal{R}_{0}^{*}$.

We prove that $u<1$ on $\mathbb{R}^{n}$. Let us set

$f(t)=\frac{\ell\,V^{\prime}(t)}{2\,\sigma}-\frac{W^{\prime}(t)}{2\,\sigma^{2}}\,,\qquad
t\in[0,1]\,,$ (6.18)

and notice that (6.15) is equivalent to $-\Delta u=f(u)$ on $\mathbb{R}^{n}$.
Since $f$ is a Lipschitz function on $[0,1]$ with $f(1)=0$, we can find $c>0$
such that $f(t)+c\,t$ is increasing on $[0,1]$, and rewrite $-\Delta u=f(u)$
as

$-\Delta\,(1-u)+c\,(1-u)=(f(t)+c\,t)\Big{|}^{t=1}_{t=u}\geq 0\,.$

We thus conclude that $v=1-u$ is non-negative on $\mathbb{R}^{n}$ and such
that $-\Delta\,v+c\,v\geq 0$. Since $v$ is non-zero (thanks to $u(x)\to 0$ as
$|x|\to\infty$), by the strong maximum principle we conclude that $v>0$ on
$\mathbb{R}^{n}$, i.e. $u<1$ on $\mathbb{R}^{n}$.

Step two: We prove statement (ii). We intend to use [PS83, Theorem 2]: if

  1. (a)

$f$ locally Lipschitz on $(0,\infty)$;

  2. (b)

$f(t)/t\to-m$ as $t\to 0^{+}$ where $m>0$;

  3. (c)

setting $F(t)=\int_{0}^{t}f(s)\,ds$, there exists $\delta>0$ such that
$F(\delta)>0$;

  4. (d)

setting $\beta=\inf\\{t>0:F(t)>0\\}$ (so that by (b) and (c),
$\beta\in(0,\delta)$), the function $t\mapsto f(t)/(t-\beta)$ is decreasing on
$(\beta,\infty)\cap\\{f>0\\}$ ;

then there is at most one $u\in C^{2}(\mathbb{R}^{n})\cap\mathcal{R}_{0}$,
with $u>0$ on $\mathbb{R}^{n}$ and $u(x)\to 0$ as $|x|\to\infty$, solving
$-\Delta u=f(u)$ on $\mathbb{R}^{n}$.

Since, by statement (i), solutions to (6.15) satisfy $0<u<1$ on
$\mathbb{R}^{n}$, in checking that $f$ as in (6.18) satisfies the above
assumptions it is only the behavior of $f$ on $(0,1)$ (and not on
$(0,\infty)$) that matters. Evidently (a) holds, since $f\in
C^{1,\alpha}[0,1]$ for some $\alpha\in(0,1)$. Assumption (b) holds with
$m=W^{\prime\prime}(0)/2\,\sigma^{2}$. Property (c) holds (with
$\delta\in(0,1)$) since

$F(t)=\int_{0}^{t}\,f(s)\,ds=\frac{\ell\,V(t)}{2\,\sigma}-\frac{W(t)}{2\,\sigma^{2}}\,,\qquad
t\in[0,1]\,,$

and $F(1)=(\ell\,V(1)/2\,\sigma)=\ell/2\,\sigma>0$ by $\ell>0$ and $W(1)=0$.
We finally prove (d). Notice that, clearly, $\beta\in(0,1)$ and, by continuity
of $F$, $F(\beta)=0$, so that, taking (A.3) and (A.6) into account, and using
$\sigma\,\ell<\nu_{0}$ and $V(1)=1$,

$\frac{\min\\{\beta^{2},(1-\beta)^{2}\\}}{C}\leq
W(\beta)=\sigma\,\ell\,V(\beta)\leq\nu_{0}\,.$ (6.19)

If $\nu_{0}<1$, then by (A.6) and (A.11) we find

$2\,\sigma^{2}\,F(t)=\sigma\,\ell\,V(t)-W(t)\leq V(t)-W(t)\leq
C\,t^{2n/(n-1)}-\frac{t^{2}}{C}<0\,,\qquad\forall t\in(0,\delta_{0})\,.$
(6.20)

By (6.20) it must be $\beta\geq\delta_{0}$. Hence, by (6.19), if $\nu_{0}$ is
sufficiently small, then $(1-\beta)^{2}\leq C\,\nu_{0}$. Up to further
decrease the value of $\nu_{0}$, we can finally entail that
$(\beta,1)\subset(1-\delta_{0},1)$, with $\delta_{0}$ as in section A.3.

We are now going to check property (d) by showing that

$f^{\prime}(t)\,(t-\beta)\leq f(t)\qquad\forall t\in(\beta,1)\,,$ (6.21)

(recall that $0<u<1$ on $\mathbb{R}^{n}$, so we can use a version of [PS83,
Theorem 2] localized to $(0,1)$). Using the explicit formula for $f$, (6.21)
is equivalent to

$\sigma\,\ell\,V^{\prime\prime}(t)\,(t-\beta)\leq\sigma\,\ell\,V^{\prime}(t)-W^{\prime}(t)+W^{\prime\prime}(t)\,(t-\beta)\,,\qquad\forall
t\in(\beta,1)\,.$ (6.22)

By (A.6), we have $W^{\prime\prime}(t)(t-\beta)>0$ on
$(\beta,1)\subset(1-\delta_{0},1)$, and since $V^{\prime}\geq 0$ on $[0,1]$,
(6.22) is implied by checking that, for every $t\in(\beta,1)$,

$\displaystyle-W^{\prime}(t)$ $\displaystyle\geq$
$\displaystyle\sigma\,\ell\,V^{\prime\prime}(t)$ $\displaystyle=$
$\displaystyle\sigma\,\ell\,\Big{\\{}\frac{n}{(n-1)^{2}}\,\frac{W(t)}{\Big{(}\int_{0}^{t}\sqrt{W}\Big{)}^{(n-2)/(n-1)}}+\frac{n}{n-1}\,\Big{(}\int_{0}^{t}\sqrt{W}\Big{)}^{1/(n-1)}\,\frac{W^{\prime}(t)}{2\,\sqrt{W(t)}}\Big{\\}}\,.$

In turn, since $W^{\prime}<0$ on $(1-\delta_{0},1)$ and
$\sigma\,\ell<\nu_{0}<1$, it is actually enough to check that

$\displaystyle-W^{\prime}(t)\geq\frac{n}{(n-1)^{2}}\,\frac{W(t)}{\Big{(}\int_{0}^{t}\sqrt{W}\Big{)}^{(n-2)/(n-1)}}\,,\qquad\forall
t\in(1-\delta_{0},1)\,.$ (6.23)

But, up to further decreasing the value of $\delta_{0}$, this is obvious:
indeed (A.6) gives $-W^{\prime}(t)\geq(1-t)/C$ and $W(t)\leq C\,(1-t)^{2}$ for
every $t\in(1-\delta_{0},1)$. ∎

###  6.3. Proof of Theorem 1.1

Theorem 2.1, Corollary 4.2, Theorem 6.1 and a scaling argument show the
validity of statements (i) and (ii), while statement (iii) follows similarly
by scaling and by Theorem 5.1. To prove the Alexandrov-type theorem, that is,
statement (iv)555Notice that we are using $\ell$ in (6.24) rather than
$\lambda$ (as done in (1.22)) to denote the Lagrange multiplier of $u$. This
is meant to avoid confusion with the function
$\lambda(\varepsilon)=(\partial\Psi/\partial m)(\varepsilon,1)$ appearing in
the argument. we consider $u\in C^{2}(\mathbb{R}^{n};[0,1])$, with $u(x)\to 0$
as $|x|\to\infty$, and solving

$-2\,\sigma^{2}\,\Delta
u=\sigma\,\ell\,V^{\prime}(u)-W^{\prime}(u)\qquad\mbox{on
$\mathbb{R}^{n}$}\,,$ (6.24)

for some $\sigma$ and $\ell$ with $0<\sigma\,\ell<\nu_{0}$. By Theorem
6.2–(i), $u\in\mathcal{R}_{0}^{*}$, and by Theorem 6.2, provided $\nu_{0}$ is
small enough, we know that there is at most one radial solution to (6.24).
Since we know that $u_{\sigma,m}$ is a radial solution of (6.24) with
$\ell=\Lambda(\sigma,m)$, we are left to prove that for every
$\ell\in(0,\nu_{0}/\sigma)$ there exists a unique
$m\in((\sigma/\varepsilon_{0})^{n},\infty)$ such that
$\Lambda(\sigma,m)=\ell$.

To this end, we first notice that, by (4.7) and by scaling, for every
$\sigma>0$ we have

$\Lambda(\sigma,m)=\frac{1}{m^{1/n}}\,\lambda\big{(}\sigma/m^{1/n})\to
0^{+}\qquad\mbox{as $m\to+\infty$}\,.$

In particular, since, by Theorem 6.1, $\Lambda(\sigma,\cdot)$ is continuous
and strictly decreasing on $((\sigma/\varepsilon_{0})^{n},\infty)$, we have
that

$\Big{\\{}\Lambda(\sigma,m):m>\Big{(}\frac{\sigma}{\varepsilon_{0}}\Big{)}^{n}\Big{\\}}=\Big{(}0,\Lambda\big{(}\sigma,(\sigma/\varepsilon_{0})^{n}\big{)}\Big{)}\,.$

Now, setting $m=(\sigma/\varepsilon_{0})^{n}$ in (6.1), that is, in

$\Big{|}m^{1/n}\,\Lambda(\sigma,m)-2\,(n-1)\,\omega_{n}^{1/n}\Big{|}\leq
C\,\frac{\sigma}{m^{1/n}}\,,$

we find that

$\Big{|}\sigma\,\Lambda\big{(}\sigma,(\sigma/\varepsilon_{0})^{n}\big{)}-2\,(n-1)\,\omega_{n}^{1/n}\,\varepsilon_{0}\Big{|}\leq
C\,\varepsilon_{0}^{2}\,,$

which implies

$\Lambda\big{(}\sigma,(\sigma/\varepsilon_{0})^{n}\big{)}\geq\frac{(n-1)\,\omega_{n}^{1/n}\,\varepsilon_{0}}{\sigma}\,,\qquad\forall\sigma>0\,,$

provided $\varepsilon_{0}$ is small enough. Up to further decrease the value
of $\nu_{0}$ so to have $\nu_{0}\leq(n-1)\,\omega_{n}^{1/n}\,\varepsilon_{0}$,
we have proved that

$(0,\nu_{0}/\sigma)\subset\Big{\\{}\Lambda(\sigma,m):m>\Big{(}\frac{\sigma}{\varepsilon_{0}}\Big{)}^{n}\Big{\\}}\,,$

and that for each $\ell\in(0,\nu_{0}/\sigma)$ there is a unique
$m>(\sigma/\varepsilon_{0})^{n}$ such that $\ell=\Lambda(\sigma,m)$, as
claimed. This completes the proof of Theorem 1.1.

##  Appendix A Frequently used auxiliary facts

###  A.1. Scaling identities

If $u\in H^{1}(\mathbb{R}^{n};[0,\infty))$, $t>0$, we set

$\rho_{t}\,u(x)=u(t^{1/n}\,x)\,,\qquad x\in\mathbb{R}^{n}\,,$

and notice that

$\displaystyle\int_{\mathbb{R}^{n}}f(\rho_{t}u)$ $\displaystyle=$
$\displaystyle\frac{1}{t}\,\int_{\mathbb{R}^{n}}f(u)\,,\qquad\int_{\mathbb{R}^{n}}|\nabla(\rho_{t}u)|^{2}=t^{(2/n)-1}\,\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,,$ (A.1) $\displaystyle\mathcal{AC}_{\varepsilon}(\rho_{t}u)$
$\displaystyle=$
$\displaystyle\varepsilon\,t^{(2/n)-1}\,\int_{\mathbb{R}^{n}}|\nabla
u|^{2}+\frac{1}{\varepsilon\,t}\int_{\mathbb{R}^{n}}W(u)=\frac{\mathcal{AC}_{\varepsilon\,t^{1/n}}(u)}{t^{(n-1)/n}}\,.$
(A.2)

whenever $f:\mathbb{R}\to\mathbb{R}$ is continuous.

###  A.2. Concentration-compactness principle

Denoting by $B_{r}(x)$ the ball of center $x$ and radius $r$ in
$\mathbb{R}^{n}$, and setting $B_{r}=B_{r}(0)$ when $x=0$, we provide a
reference statement for Lions’ concentration-compactness criterion, which is
repeatedly used in our arguments: if $\\{\mu_{j}\\}_{j}$ is a sequence of
probability measures in $\mathbb{R}^{n}$, then, up to extract subsequences and
compose each $\mu_{j}$ with a translation, one the following mutually
excluding possibilities holds:

Compactness case: for every $\tau>0$ there exists $R>0$ such that

$\inf_{j}\,\mu_{j}(B_{R})\geq 1-\tau\,;$

Vanishing case: for every $R>0$,

$\lim_{j\to\infty}\,\sup_{x\in\mathbb{R}^{n}}\mu_{j}(B_{R}(x))=0\,;$

Dichotomy case: there exists $\alpha\in(0,1)$ such that for every $\tau>0$ one
can find $S>0$ with $S_{j}\to\infty$ such that

$\sup_{j}\,|\alpha-\mu_{j}(B_{S})|<\tau\,,\qquad\sup_{j}\,\big{|}(1-\alpha)-\mu_{j}(\mathbb{R}^{n}\setminus
B_{S_{j}})\big{|}<\tau\,.$

Notice that the formulation of the dichotomy case used here is a bit more
descriptive than the original one presented in [Lio84, Lemma I]. Its validity
is inferred by a quick inspection of the proof presented in the cited
reference.

###  A.3. Estimates for $W$, $\Phi$ and $V$

Throughout the paper we work with a double well potential $W\in C^{2,1}[0,1]$
satisfying (1.11) and (1.12), that is,

$\displaystyle\mbox{$W(0)=W(1)=0$}\,,\quad\mbox{$W>0$ on
$(0,1)$}\,,\quad\mbox{$W^{\prime\prime}(0),W^{\prime\prime}(1)>0$}\,,$ (A.3)
$\displaystyle\int_{0}^{1}\sqrt{W}=1\,.$ (A.4)

Frequently used properties of $W$ are the validity, for a universal constant
$C$, of the expansion

$\Big{|}W(b)-W(a)-W^{\prime}(a)(b-a)-W^{\prime\prime}(a)\frac{(b-a)^{2}}{2}\Big{|}\leq
C\,|b-a|^{3}\,,\qquad\forall a,b\in[0,1]\,,$ (A.5)

and the existence of a universal constant $\delta_{0}<1/2$ such that

$\begin{split}&\frac{1}{C}\leq\frac{W}{t^{2}}\,,\frac{W^{\prime}}{t}\,,W^{\prime\prime}\leq
C\quad\hskip 42.67912pt\mbox{on $(0,\delta_{0}]$}\,,\\\
&\frac{1}{C}\leq\frac{W}{(1-t)^{2}}\,,\frac{-W^{\prime}}{1-t}\,,W^{\prime\prime}\leq
C\quad\hskip 11.38092pt\mbox{on $[1-\delta_{0},1)$}\,.\end{split}$ (A.6)

We can use (A.6) to quantify the behaviors near the wells of $\Phi$ and,
crucially, of $V$. We first notice that, by (A.3), $\Phi\in C^{3}_{{\rm
loc}}(0,1)$, with

$\Phi^{\prime}=\sqrt{W}\,,\quad\Phi^{\prime\prime}=\frac{W^{\prime}}{2\,\sqrt{W}}\,,\quad\Phi^{\prime\prime\prime}=\frac{W^{\prime\prime}}{2\,\sqrt{W}}-\frac{(W^{\prime})^{2}}{4\,W^{3/2}}\,,\qquad\mbox{on
$(0,1)$}\,.$

By (A.6) and (A.4) we thus see that $\Phi$ satisfies

$\begin{split}&\frac{1}{C}\leq\frac{\Phi}{t^{2}}\,,\frac{\Phi^{\prime}}{t}\,,\Phi^{\prime\prime}\leq
C\,,\qquad\mbox{on $(0,\delta_{0}]$}\,,\\\
&\frac{1}{C}\leq\frac{1-\Phi}{(1-t)^{2}}\,,\frac{\Phi^{\prime}}{1-t}\,,-\Phi^{\prime\prime}\leq
C\,,\qquad\mbox{on $[1-\delta_{0},1)$}\,,\end{split}$ (A.7)

from which we easily deduce

$|\Phi(b)-\Phi(a)|\geq\frac{(b-a)^{2}}{C}\,,\qquad\forall a,b\in[0,1]\,.$
(A.8)

Moreover, by exploiting (A.7) and setting for brevity $a=W^{\prime\prime}(0)$,
we see that as $t\to 0^{+}$

$\displaystyle\Phi^{\prime\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{2\,W^{\prime\prime}\,W-(W^{\prime})^{2}}{4\,W^{3/2}}=\frac{2\,(a+{\rm
O}(t))\,(a\,(t^{2}/2)+{\rm O}(t^{3}))-(a\,t+{\rm
O}(t^{2}))^{2}}{4\,(a\,(t^{2}/2)+{\rm O}(t^{3}))^{3/2}}$ $\displaystyle=$
$\displaystyle\frac{{\rm O}(t^{3})}{4\,a^{3/2}\,t^{3}+{\rm o}(t^{3})}\,,$

and a similar computation holds for $t\to 1^{-}$, so that

$|\Phi^{\prime\prime\prime}|\leq C\,,\qquad\mbox{on
$(0,\delta_{0})\cup(1-\delta_{0},1)$}\,.$ (A.9)

By (A.7) and (A.9) we see that $\Phi\in C^{2,1}[0,1]$ with a universal
estimate on its $C^{2,1}[0,1]$-norm: in particular,

$\Big{|}\Phi(b)-\Phi(a)-\Phi^{\prime}(a)(b-a)-\Phi^{\prime\prime}(a)\frac{(b-a)^{2}}{2}\Big{|}\leq
C\,|b-a|^{3}\,,\qquad\forall a,b\in(0,1)\,.$ (A.10)

Since $V=\Phi^{1+\alpha}$ for $\alpha=1/(n-1)\in(0,1]$ (recall that $n\geq 2$)
and $\Phi(t)=0$ if and only if $t=0$, we easily see that $V\in C^{3}_{{\rm
loc}}(0,1)$, with

$\displaystyle V^{\prime}=(1+\alpha)\,\Phi^{\alpha}\,\Phi^{\prime}\,,\qquad
V^{\prime\prime}=(1+\alpha)\Big{\\{}\alpha\,\frac{(\Phi^{\prime})^{2}}{\Phi^{1-\alpha}}+\Phi^{\alpha}\,\Phi^{\prime\prime}\Big{\\}}\,,$
$\displaystyle|V^{\prime\prime\prime}|\leq
C(\alpha)\,\Big{\\{}\frac{(\Phi^{\prime})^{3}}{\Phi^{2-\alpha}}+\frac{\Phi^{\prime}\,|\Phi^{\prime\prime}|}{\Phi^{1-\alpha}}+\Phi^{\alpha}\,|\Phi^{\prime\prime\prime}|\Big{\\}}\,.$

By (A.10), and keeping track of the sign of $\Phi^{\prime\prime}$ and of the
fact that negative powers of $\Phi(t)$ are large only near $t=0$, but are
bounded near $t=1$, we find that

$\begin{split}&\frac{1}{C}\leq\frac{V}{t^{2+2\alpha}}\,,\frac{V^{\prime}}{t^{1+2\alpha}}\,,\frac{V^{\prime\prime}}{t^{2\alpha}}\leq
C\,,\hskip
19.91684pt\quad|V^{\prime\prime\prime}|\leq\frac{C}{t^{1-2\alpha}}\qquad\hskip
5.69046pt\mbox{on $(0,\delta_{0}]$}\,,\\\
&\frac{1}{C}\leq\frac{1-V}{(1-t)^{2}}\,,\frac{V^{\prime}}{1-t}\leq
C\,,\quad|V^{\prime\prime}|\,,|V^{\prime\prime\prime}|\leq C\,,\qquad\mbox{on
$[1-\delta_{0},1)$}\,.\end{split}$ (A.11)

In particular, $V\in C^{2,\gamma(n)}[0,1]$,
$\gamma(n)=\min\\{1,2/(n-1)\\}\in(0,1]$, with second order Taylor expansions
of the form

$\Big{|}V(b)-V(a)-V^{\prime}(a)(b-a)-V^{\prime\prime}(a)\frac{(b-a)^{2}}{2}\Big{|}\leq
C\,|b-a|^{2+\gamma(n)}\,,\qquad\forall a,b\in(0,1)\,.$ (A.12)

We finally notice that we can find a universal constant $C$ such that

$\frac{t^{2}}{C}\leq W(t)\,,\qquad V(t)\leq C\,t^{2}\,,\qquad V(t)\leq
C\,W(t)\,,\qquad\forall t\in(0,1-\delta_{0})\,,$ (A.13)

(as it is easily deduced from the bounds on $W$ and $V$ in (A.6) and (A.11)
and from the fact that $W>0$ on $(0,1)$); and that we can also find $C$ so
that

$V(t)\geq\frac{1}{C}\,,\qquad\forall t\in(\delta_{0},1)\,.$ (A.14)

###  A.4. Estimates for the optimal transition profile $\eta$

A crucial object in the analysis of the Allen-Cahn energy is of course the
optimal transition profile $\eta$, defined by the first order ODE

$\left\\{\begin{split}&\eta^{\prime}=-\sqrt{W(\eta)}\qquad\mbox{on
$\mathbb{R}$}\,,\\\ &\eta(0)=\frac{1}{2}\,,\end{split}\right.$ (A.15)

which can be seen to satisfy (see, e.g. [LM16a]) $\eta\in
C^{2,1}(\mathbb{R})$, $\eta^{\prime}<0$ on $\mathbb{R}$ (and
$-C\leq\eta^{\prime}\leq-1/C$ for $|s|\leq 1$), $\eta(-\infty)=1$, and
$\eta(+\infty)=0$, with the exponential decay properties

$1-\eta(s)\leq C\,e^{s/C}\qquad\forall s<0\,,\qquad\eta(s)\leq
C\,e^{-s/C}\qquad\forall s>0\,,$ (A.16)

for a universal constant $C$. Similarly, by combining (A.16) with (A.15), with
the second order ODE satisfied by $\eta$, namely,

$2\,\eta^{\prime\prime}=W^{\prime}(\eta)\qquad\mbox{on $\mathbb{R}$}\,,$
(A.17)

and with (A.6) we see that also the first and second derivatives of $\eta$
decay exponentially

$|\eta^{\prime}(s)|\,,|\eta^{\prime\prime}(s)|\leq C\,e^{-|s|/C}\qquad\forall
s\in\mathbb{R}\,.$ (A.18)

Combining again (A.16) and (A.6) we also see that

$s\in\mathbb{R}\mapsto\,1_{(-\infty,0)}(s)-V(\eta(s-\tau))$

belongs to $L^{1}(\mathbb{R})$ for every $\tau\in\mathbb{R}$, with

$\tau\in\mathbb{R}\mapsto\int_{-\infty}^{\infty}\Big{(}1_{(-\infty,0)}(s)-V(\eta(s-\tau))\Big{)}\,ds$

increasing in $\tau$ and converging to $\mp\infty$ as $\tau\to\pm\infty$. In
particular, there is a unique universal constant $\tau_{0}$ such that

$\int_{-\infty}^{\infty}\Big{(}1_{(-\infty,0)}(s)-V(\eta(s-\tau_{0}))\Big{)}\,ds=0\,.$
(A.19)

The constant $\tau_{0}$ appears in the computation of the first order
expansion of $\psi(\varepsilon)$ as $\varepsilon\to 0^{+}$ and can be
characterized, equivalently, to be

$\tau_{0}=\int_{\mathbb{R}}\eta^{\prime}\,V^{\prime}(\eta)\,s\,ds\,.$ (A.20)

Indeed, (A.19) gives

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\Big{(}1_{(-\infty,0)}(s)-V(\eta(s-\tau_{0}))\Big{)}\,ds$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{0}\Big{(}1-V(\eta(s-\tau_{0}))\Big{)}\,ds-\int_{0}^{\infty}V(\eta(s-\tau_{0}))\,ds$
$\displaystyle=$
$\displaystyle-\int_{-\infty}^{0}\,ds\,\int_{-\infty}^{s-\tau_{0}}\eta^{\prime}(t)\,V^{\prime}(\eta(t))\,dt+\int_{0}^{\infty}ds\int_{s-\tau_{0}}^{\infty}\eta^{\prime}(t)\,V^{\prime}(\eta(t))\,dt\,.$

Both integrands are non-negative, therefore by Fubini’s theorem

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{-\tau_{0}}\,dt\,\int_{t+\tau_{0}}^{0}\,\eta^{\prime}(t)\,V^{\prime}(\eta(t))\,ds-\int_{-\tau_{0}}^{\infty}dt\int_{0}^{t+\tau_{0}}\eta^{\prime}(t)\,V^{\prime}(\eta(t))\,ds$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{-\tau_{0}}\,(t+\tau_{0})\,\eta^{\prime}(t)\,V^{\prime}(\eta(t))\,dt+\int_{-\tau_{0}}^{\infty}(t+\tau_{0})\eta^{\prime}(t)\,V^{\prime}(\eta(t))\,dt$

that is

$\int_{\mathbb{R}}\,\eta^{\prime}\,V^{\prime}(\eta)\,t\,dt=-\tau_{0}\,\int_{\mathbb{R}}\eta^{\prime}V^{\prime}(\eta)=V(1)\,\tau_{0}=\tau_{0}\,,$

as claimed.

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