# Asymptotic profile and Morse index of the radial solutions of the Hénon
equation

Wendel Leite da Silva and Ederson Moreira dos Santos  Instituto de Ciências
Matemáticas e de Computação
Universidade de São Paulo, CEP 13560-970 - São Carlos - SP - Brazil
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We consider the Hénon equation

$-\Delta u=|x|^{\alpha}|u|^{p-1}u\ \ \textrm{in}\ \ B^{N},\quad u=0\ \
\textrm{on}\ \ \partial B^{N},$ ($P_{\alpha}$)

where $B^{N}\subset\mathbb{R}^{N}$ is the open unit ball centered at the
origin, $N\geq 3$, $p>1$ and $\alpha>0$ is a parameter. We show that, after a
suitable rescaling, the two-dimensional Lane-Emden equation

$-\Delta w=|w|^{p-1}w\quad\text{in}\ B^{2},\quad w=0\quad\text{on}\ \partial
B^{2},$

where $B^{2}\subset\mathbb{R}^{2}$ is the open unit ball, is the limit problem
of ($P_{\alpha}$), as $\alpha\to\infty$, in the framework of radial solutions.
We exploit this fact to prove several qualitative results on the radial
solutions of ($P_{\alpha}$) with any fixed number of nodal sets: asymptotic
estimates on the Morse indices along with their monotonicity with respect to
$\alpha$; asymptotic convergence of their zeros; blow up of the local extrema
and on compact sets of $B^{N}$. All these results are proved for both positive
and nodal solutions.

###### Key words and phrases:

Semilinear elliptic equations; Hénon equation; Radial solutions; Qualitative
properties

###### 2010 Mathematics Subject Classification:

35B06; 35B40; 35J15; 35J61

Wendel Leite da Silva was partially supported by CNPq and CAPES. Ederson
Moreira dos Santos was partially supported by CNPq grant 307358/2015-1 and
FAPESP grant 2015/17096-6.

##  1\. Introduction

The qualitative analysis of solutions of partial differential equations is a
very important field of research, and it turns out that the Hénon equation
[15] is an excellent prototype for the study of some fundamental problems in
this subject. For example, the symmetry of least energy solutions and least
energy nodal solutions [5, 20, 23, 24] and some concentration phenomena [8, 7,
11]. To investigate the symmetry of least energy solutions and least energy
nodal solutions, which are known as low Morse indices solutions, a very useful
argument, based on some ideas introduced in [1], is that radially symmetric
solutions have large Morse indices. However, the results in [1] are presented
for autonomous problems and their proofs cannot be adapted to the
nonautonomous case, which includes the Hénon equation. Indeed, a conjecture on
the symmetry breaking of the least energy nodal solutions of the superlinear
subcritical Hénon equation remained opened for at least 14 years, since the
paper [5], was partially solved in [20] for the case of $N=2$ (see also [18]),
and completely settled very recently in [3, 4]. As presented ahead, as a
byproduct of the main theorems in this paper, we complement some of the
results in [2, 3, 4, 6, 18, 19, 20] with new information on the asymptotic
profile and Morse indices of the radial solutions of the Hénon equation, for
both positive and nodal solutions.

We consider the Hénon equation

$\left\\{\begin{array}[]{l}\begin{aligned} -\Delta
u&=|x|^{\alpha}|u|^{p-1}u&\textrm{in}&\ \ B^{N},\vspace{0.3 cm}\\\
u&=0&\textrm{on}&\ \ \partial B^{N},\end{aligned}\end{array}\right.$
($P_{\alpha}$)

where $B^{N}\subset\mathbb{R}^{N}$ is the open unit ball centered at the
origin, $N\geq 3$, $p>1$ and $\alpha>0$ is a parameter.

We recall that, for each positive integer $m$, it is shown in [21] that
($P_{\alpha}$) admits a unique radial $C^{2}(\,\overline{B^{N}}\,)$-solution,
positive at zero, with exactly $m$ nodal sets, for $1<p<2^{*}_{\alpha}-1$,
where $2^{*}_{\alpha}:=2(N+\alpha)/(N-2)$. For a fixed positive integer $m$,
we denote this solution by $u_{\alpha}$. Observe that the condition
$p<2^{*}_{\alpha}-1$ with $\alpha>0$ is equivalent to $\alpha>\alpha_{p}$,
where

$\alpha_{p}:=\max\left\\{0,\frac{p(N-2)-(N+2)}{2}\right\\}.$

We show that, after a suitable rescaling, the two-dimensional Lane-Emden
equation

$-\Delta w=|w|^{p-1}w\quad\text{in}\ B^{2},\quad w=0\quad\text{on}\ \partial
B^{2},$ ($L$)

where $B^{2}\subset\mathbb{R}^{2}$ is the open unit ball, is the limit problem
of ($P_{\alpha}$), as $\alpha\to\infty$, in the framework for radial
solutions.

From now on, for $m\in\mathbb{N}$ and $p>1$ fixed, we denote by $w$ the unique
radial solution of ($L$) with exactly $m$ nodal sets and such that $w(0)>0$;
see [22, 16]. We also denote by $\lambda_{1}<\ldots<\lambda_{m}<0$ the radial
negative eigenvalues of the singular problem

$-\Delta\psi-p|w|^{p-1}\psi=\lambda\frac{\psi}{|y|^{2}}\ \ \text{in}\ \
B^{2}\backslash\\{0\\},\quad\psi=0\ \ \text{on}\ \ \partial B^{2};$

see [14, Proposition 2.9] and [3, Proposition 1.1]. To simplify notation, we
identify radial functions defined on $\overline{B^{N}}$ with their
representative defined on the interval $[0,1]$.

######  Theorem 1.1.

For $m\in\mathbb{N}$ and $p>1$ fixed, let $u_{\alpha}$ and $w$ be the radial
solutions of ($P_{\alpha}$) and ($L$), respectively, with $m$ nodal sets such
that $u_{\alpha}(0)>0$ and $w(0)>0$. Then

$v_{\alpha}(t):=\left(\frac{2}{\alpha+2}\right)^{\frac{2}{p-1}}u_{\alpha}\left(t^{\frac{2}{\alpha+2}}\right),\quad
t\in[0,1]$ (1.1)

converges to $w$ in $C^{1}([0,1])\cap C^{2}([\varepsilon,1])$ as
$\alpha\to\infty$, for all $\varepsilon\in(0,1)$.

The change of variables (1.1) was used in [9]; see also [13, 18, 20]. There
the authors show that, in the particular case of $N=2$, for all $\alpha>0$,
the function $v_{\alpha}$ is precisely the radial solution of ($L$) with $m$
nodal sets. In other words, the sequence of functions
$(v_{\alpha})_{\alpha>0}$ is constant if $N=2$. Although not constant for
$N\geq 3$, Theorem 1.1 shows that the sequence (1.1) has the same limit for
every $N\geq 2$.

######  Remark 1.2.

To prove Theorem 1.1 we show that the absolute values of all the local extrema
of $u_{\alpha}$ explode at the same rate $\alpha^{2/(p-1)}$ as
$\alpha\to\infty$; see Theorem 1.6 (iii) ahead, and [7, Theorem 3.1-E] for the
positive radial solution. For instance, this rate also holds for the
$L^{\infty}$-norm of the least energy solutions of ($P_{\alpha}$); see [8,
Lemma 4.3] and [7, Theorem 4.1-E].

Based on Theorem 1.1, we obtain some information on the Morse index of the
solution $u_{\alpha}$, as $\alpha\to\infty$. Here the symbols $\lceil.\rceil$
and $\lfloor.\rfloor$ represent, respectively, the ceiling and floor functions
whose definitions are given as
$\lceil\beta\rceil:=\min\\{k\in\mathbb{Z}:k\geq\beta\\}$,
$\lfloor\beta\rfloor:=\max\\{k\in\mathbb{Z}:k\leq\beta\\}$, for
$\beta\in\mathbb{R}$.

######  Theorem 1.3.

For $m\in\mathbb{N}$ and $p>1$ fixed, let $m(u_{\alpha})$ be the Morse index
of a radial solution $u_{\alpha}$ of ($P_{\alpha}$) with $m$ nodal sets. Then
there exists $\alpha^{*}=\alpha^{*}(p,N,m)>\alpha_{p}$, that does not depend
on $\alpha$, such that

$m(u_{\alpha})\geq
m+m\sum_{j=1}^{J_{\alpha}}N_{j}\quad\forall\alpha\geq\alpha^{*},$ (1.2)

where

$J_{\alpha}:=\left\lceil\frac{\sqrt{(N-2)^{2}-(\lambda_{m}/2)\alpha^{2}}}{2}\right\rceil\quad\text{and}\quad
N_{j}:=\frac{(N+2j-2)(N+j-3)!}{(N-2)!j!}.$

Moreover, if $m\geq 2$, then for each $\theta>1$, there exists
$\alpha^{\star}=\alpha^{\star}(\theta,p,N,m)>\alpha_{p}$ independent of
$\alpha$ such that

$m(u_{\alpha})\geq
m+(m-1)\sum_{j=1}^{K_{\alpha}(\theta)}N_{j}\quad\forall\alpha\geq\alpha^{\star},$
(1.3)

where

$K_{\alpha}(\theta):=\left\lceil\frac{\sqrt{(N-2)^{2}-(\lambda_{m-1}/\theta)\alpha^{2}}}{2}\right\rceil.$

######  Remark 1.4.

Theorem 1.3 deserves some comments. Comparing to [4, Theorem 1.1], our
contribution here is twofold. Firstly, our result is also valid for positive
solutions ($m=1$), while [4] gives no new information for positive solutions.
Secondly, for nodal solutions, we improve the lower bounds for the Morse
indices for large values of $\alpha$. Indeed, it is shown in [4, Theorem 1.1]
that

$m(u_{\alpha})\geq
m+(m-1)\sum_{j=1}^{1+\lfloor\frac{\alpha}{2}\rfloor}N_{j}\quad\forall\alpha>\alpha_{p}.$

Moreover, from [10, Eq. (6.11)] (see also [4, Proposition 3.3]), it is known
that $-\lambda_{m-1}>1$ and hence for fixed $\theta_{0}>1$ close enough to
$1$, one has that $-\lambda_{m-1}/\theta_{0}>1$ and thus
$K_{\alpha}(\theta_{0})>1+\left\lfloor\frac{\alpha}{2}\right\rfloor$ for all
$\alpha$ large enough. Theorem 1.3 also presents an improvement on the results
[19, Theorem 1.1 i)], once we have an explicit lower bound. Finally, for nodal
solutions and large values of $\alpha$, it is clear that the lower bound in
(1.3) is much better than that in (1.2), where the latter is also valid for
positive solutions ($m=1$).

In [18] we proved that, in dimension $N=2$, the Morse index of the radial
solutions of ($P_{\alpha}$) with the same number of nodal sets is monotone
nondecreasing with respect to $\alpha$. The case of $N=2$ is special, we took
advantage of a change of variables that relates Hénon equations with different
weights. Although such transformation is not available for dimensions higher
than two, here we prove a similar monotonicity result for $N\geq 3$ and large
values of $\alpha$.

######  Theorem 1.5.

Let $p>1$ and $m\in\mathbb{N}$. Let $u_{\alpha}$ and $u_{\beta}$ be radial
solutions of ($P_{\alpha}$) and $(P_{\beta})$, respectively, with $m$ nodal
sets. Then exists $\alpha_{*}>\alpha_{p}$ such that

$m(u_{\alpha})\leq
m(u_{\beta}),\quad\forall\alpha,\beta\in[\alpha_{*},\infty),\ \alpha<\beta.$

Next, we present a result on the asymptotic concentration of zeros and blow up
of radial solutions of ($P_{\alpha}$), as $\alpha\to\infty$. If $m=1$, then
$x=0$ is the unique local extremum of $u_{\alpha}$. If $m\geq 2$, denote by
$r_{1,\alpha}<\cdots<r_{m,\alpha}=1$ the zeros of $u_{\alpha}$ in $(0,1]$ and
set

$\mathcal{M}_{1,\alpha}:=\max\\{u_{\alpha}(r):0\leq r\leq
r_{1,\alpha}\\},\quad\mathcal{M}_{i+1,\alpha}:=\max\\{|u_{\alpha}(r)|:r_{i,\alpha}\leq
r\leq r_{i+1,\alpha}\\}$

for $i=1,\cdots,m-1$.

######  Theorem 1.6.

For $m\in\mathbb{N}$ and $p>1$ fixed, let $u_{\alpha}$ and $w$ be the radial
solutions of ($P_{\alpha}$) and ($L$), respectively, with $m$ nodal sets such
that $u_{\alpha}(0)>0$ and $w(0)>0$.

  1. (i)

For each $i=1,\cdots,m$,

$\alpha(1-r_{i,\alpha})\to-2\log(t_{i})\quad\text{as}\ \alpha\to\infty,$

where $r_{1,\alpha}<\ldots<r_{m-1,\alpha}<r_{m,\alpha}=1$ and
$t_{1}<\ldots<t_{m-1}<t_{m}=1$ are the zeros of $u_{\alpha}$ and $w$,
respectively.

  2. (ii)

Let $0<R_{0}<1$. Then

$\left(\frac{2}{\alpha+2}\right)^{\frac{2}{p-1}}u_{\alpha}(x)\to w(0)$

uniformly in $B(0,R_{0})=\\{x\in\mathbb{R}^{N};|x|<R_{0}\\}$ as
$\alpha\to\infty$.

  3. (iii)

There exist $\alpha_{0}>\alpha_{p}$ and constants $C_{1},C_{2}>0$ such that

$C_{1}\left(\frac{\alpha+2}{2}\right)^{\frac{2}{p-1}}\leq\mathcal{M}_{i,\alpha}\leq
C_{2}\left(\frac{\alpha+2}{2}\right)^{\frac{2}{p-1}}$

for all $i=1,\cdots,m$ and $\alpha\geq\alpha_{0}$.

######  Remark 1.7.

We make some comments on Theorem 1.6.

  1. (i)

Regarding Theorem 1.6(i), in the case with $\alpha>0$ fixed and $p\nearrow
2^{*}_{\alpha}-1$, a different phenomenon occurs. In [2], the authors studied
the asymptotic behavior of radial solutions of ($P_{\alpha}$) for $\alpha>0$
fixed and $p\nearrow 2^{*}_{\alpha}-1$. In this case, if
$r_{1,p}<\ldots<r_{m,p}=1$ are the zeros of $u_{p}$, the unique radial nodal
solutions of ($P_{\alpha}$) with $m$ nodal sets and $u_{p}(0)>0$, then
$r_{i,p}\to 0$ as $p\nearrow 2^{*}_{\alpha}-1$, for all $i=1,\ldots,m-1$; see
[2, Theorem 1.2].

  2. (ii)

Let $\omega_{\alpha}$ be the least energy solution of the Hénon equation
($P_{\alpha}$) and $R_{0}\in(0,1)$. It was proved in [6, Eq. (32)] that

$\left(\frac{2}{\alpha+2}\right)^{\frac{2}{p-1}}\omega_{\alpha}(x)\to 0$

uniformly in $B(0,R_{0})$, as $\alpha\to\infty$. According to Theorem 1.6(ii),
this is in contrast with the radial, either nodal or positive, solutions of
($P_{\alpha}$).

For $p>1$ and $\alpha>\alpha_{p}$, denote by $S_{\alpha,p}^{R}$ the best
constant of the embedding $H^{1}_{0,rad}(B^{N})\subset
L^{p+1}(B^{N},|x|^{\alpha})$, that is

$S_{\alpha,p}^{R}:=\inf_{0\neq u\in
H^{1}_{0,rad}(B^{N})}\displaystyle{\frac{\int_{B^{N}}|\nabla
u|^{2}}{\left(\int_{B^{N}}|x|^{\alpha}|u|^{p+1}\right)^{\frac{2}{p+1}}}}.$

Actually, the condition $\alpha>\alpha_{p}$ implies that the above embedding
is compact, and hence $S_{\alpha,p}^{R}$ is achieved by a positive radial
function $u_{\alpha}$, which is also, up to scaling, the positive radial
solution of ($P_{\alpha}$). It is proved in [24, Theorem 4.1] that

$\lim_{\alpha\to\infty}\left(\frac{N}{\alpha+N}\right)^{\frac{p+3}{p+1}}S_{\alpha,p}^{R}=C\in(0,\infty),$

Here, we give a precise characterization for the constant $C$ above. Let
$S_{p}$ be the best constant of the embedding $H^{1}_{0}(B^{2})\subset
L^{p+1}(B^{2})$, namely

$S_{p}:=\inf_{0\neq v\in
H^{1}_{0}(B^{2})}\displaystyle{\frac{\int_{B^{2}}|\nabla
v|^{2}}{\left(\int_{B^{2}}|v|^{p+1}\right)^{\frac{2}{p+1}}}}.$

######  Theorem 1.8.

For any $p>1$,

$\lim_{\alpha\to\infty}\left(\frac{2}{\alpha+2}\right)^{\frac{p+3}{p+1}}S_{\alpha,p}^{R}=\left(\frac{\omega_{N-1}}{2\pi}\right)^{\frac{p-1}{p+1}}S_{p}.$

The rest of this paper is organized as follows. In Section 2 we prove sharp
estimates for the energy of the solution $u_{\alpha}$, which are used in
Section 3 to prove the asymptotic behavior of $\|u_{\alpha}\|_{\infty}$, as
$\alpha\to\infty$. Section 4 is devoted to estimates on the local extrema of
$u_{\alpha}$, in particular the proof of Theorem 1.6 (iii). In Section 5 we
prove Theorems 1.1 and 1.8 and conclude the proof of Theorem 1.6. Finally,
Section 6 is devoted to the analysis of the asymptotic distribution for the
spectrum of some linearized operators and proofs of Theorems 1.3 and 1.5.

##  2\. Energy levels and their asymptotic estimates

Set

$\varphi_{\alpha}(u):=\frac{1}{2}\int_{B^{N}}|\nabla
u|^{2}dx-\frac{1}{p+1}\int_{B^{N}}|x|^{\alpha}|u|^{p+1}dx.$

Since the embedding $H^{1}_{0,rad}(B^{N})\subset L^{p+1}(B^{N},|x|^{\alpha})$
is compact for all $1<p<2^{*}_{\alpha}-1$, $\varphi_{\alpha}$ is well defined
in $H^{1}_{0,rad}(B^{N})$ and the radial solutions of ($P_{\alpha}$) are
exactly the critical points of $\varphi_{\alpha}$. In particular, these
solutions lie on the Nehari manifold

$\mathcal{N}^{\alpha}:=\left\\{0\neq u\in
H^{1}_{0,rad}(B^{N}):\varphi_{\alpha}^{\prime}(u)u=0\right\\}=\left\\{0\neq
u\in H^{1}_{0,rad}(B^{N}):\int_{B^{N}}|\nabla
u|^{2}=\int_{B^{N}}|x|^{\alpha}|u|^{p+1}\right\\}.$

We set

$H_{m}:=\\{u\in H^{1}_{0,rad}(B^{N}):\ u\ \text{has precisely}\ m\ \text{nodal
sets}\\}.$

For $u\in H_{m}$, let $R_{1},\cdots,R_{m}$ be the nodal regions of $u$. We
introduce the function $u_{i}$ defined by

$u_{i}(x):=\left\\{\begin{array}[]{l}\begin{aligned} &u(x)\ &\text{if}&\ x\in
R_{i}\\\ &0\ &\text{if}&\ x\in B^{N}\backslash
R_{i}.\end{aligned}\end{array}\right.$

Since $u_{\alpha}$ and $-u_{\alpha}$ are the radial solutions of
($P_{\alpha}$) with precisely $m$ nodal sets, we can define the level of
radial solutions in $H_{m}$ as

${C}_{\alpha,m}:=\varphi_{\alpha}(u_{\alpha}).$

We also define

$\overline{{C}}_{\alpha,m}:=\inf\\{\varphi_{\alpha}(u):u\in H_{m}\ \text{and}\
u_{i}\in\mathcal{N}^{\alpha}\ \forall i=1,\cdots,m\\}$

and

$\widetilde{{C}}_{\alpha,m}:=\inf_{u\in
H_{m}}\max_{t\in\mathbb{R}^{m}_{+}}\sum_{i=1}^{m}\varphi_{\alpha}(t_{i}u_{i}),$

where here
$\mathbb{R}^{m}_{+}:=\\{t=(t_{1},\cdots,t_{m})\in\mathbb{R}^{m}:t_{i}>0\
\forall i=1,\cdots,m\\}$.

######  Proposition 2.1.

${C}_{\alpha,m}=\overline{{C}}_{\alpha,m}=\widetilde{{C}}_{\alpha,m}$.

To prove this proposition, we recall the procedure to produce a radial
solution of ($P_{\alpha}$) with $m$ nodal sets, provided $\alpha>\alpha_{p}$.
The compactness of the embedding $H^{1}_{0,rad}(B^{N})\subset
L^{p+1}(B^{N},|x|^{\alpha})$ implies that the infimum of $\varphi_{\alpha}$ on
$\mathcal{N}^{\alpha}$ is achieved at the positive radial solution of
($P_{\alpha}$). When $m\geq 2$, a radial solution of ($P_{\alpha}$) with $m$
nodal sets can be obtained by the Nehari method; [25, Chapter 4]. For that, we
introduce the spaces

$\displaystyle X_{s,t}$ $\displaystyle:=\\{u\in H^{1}_{0,rad}(B^{N}):u(r)=0\
\forall r\in[0,s]\cup[t,1]\\},\quad$ $\displaystyle 0<s<t\leq 1,$
$\displaystyle X_{0,t}$ $\displaystyle:=\\{u\in H^{1}_{0,rad}(B^{N}):u(r)=0\
\forall r\in[t,1]\\},\quad$ $\displaystyle 0<t<1,$

and the Nehari sets

$\mathcal{N}^{\alpha}_{s,t}:=\left\\{0\neq u\in X_{s,t}:\int_{B^{N}}|\nabla
u|^{2}dx=\int_{B^{N}}|x|^{\alpha}|u|^{p+1}dx\right\\}$

for $0\leq s<t\leq 1$, and solve the minimization problem

$\widehat{{C}}_{\alpha,m}:=\inf\left\\{\sum_{i=1}^{m}\min_{\mathcal{N}^{\alpha}(t_{i-1},t_{i})}\varphi_{\alpha}:0=t_{0}<t_{1}<\cdots<t_{m}=1\right\\}.$
(2.1)

Arguing as in [25, Proposition 4.4-(d)], $\widehat{{C}}_{\alpha,m}$ is
achieved, say at $0=r_{0}<r_{1}<\cdots<r_{m}=1$. Moreover, for each
$i=1,\cdots,m$, there exists a nonnegative function $u_{i}\in
X_{r_{i-1},r_{i}}$ such that

$\varphi_{\alpha}(u_{i})=\min_{\mathcal{N}^{\alpha}_{r_{i-1},r_{i}}}\varphi_{\alpha}.$

It can then be shown, like in [25, Lemma 4.7], that the function
$u:B^{N}\to\mathbb{R}$ given by $u(x)=(-1)^{i-1}u_{i}(x)$ if
$|x|\in[r_{i-1},r_{i})$ is of class $C^{2}$ and therefore it is the radial
solution of ($P_{\alpha}$) with exactly $m$ nodal sets such that $u(0)>0$.
Consequently, $u=u_{\alpha}$ and

${C}_{\alpha,m}=\varphi_{\alpha}(u_{\alpha})=\widehat{{C}}_{\alpha,m}.$ (2.2)

Now we can prove Proposition 2.1.

###### Proof of Proposition 2.1.

Since $u_{\alpha}\in H_{m}$ and the restrictions of $u_{\alpha}$ to its nodal
regions lie on $\mathcal{N}^{\alpha}$, we have
$\overline{{C}}_{\alpha,m}\leq{C}_{\alpha,m}$. By contradiction, suppose
$\overline{{C}}_{\alpha,m}<{C}_{\alpha,m}$. Then, by (2.2), there exists $u\in
H_{m}$ with $u_{i}\in\mathcal{N}^{\alpha}$ for $i=1,\cdots,m$ such that
$\varphi_{\alpha}(u)<\widehat{{C}}_{\alpha,m}$. Hence, if
$s_{1}<\cdots<s_{m}=1$ are the zeros of $u$ in $(0,1]$ and $0=s_{0}$, then

$\widehat{{C}}_{\alpha,m}>\varphi_{\alpha}(u)=\sum_{i=1}^{m}\varphi_{\alpha}(u_{i})\geq\sum_{i=1}^{m}\min_{\mathcal{N}^{\alpha}(s_{i-1},s_{i})}\varphi_{\alpha}\geq\widehat{{C}}_{\alpha,m},$

which is a contradiction.

To check the second equality, it suffices to observe that
$\displaystyle\max_{t\in\mathbb{R}^{m}_{+}}\displaystyle\sum_{i=1}^{m}\varphi_{\alpha}(t_{i}u_{i})$
is achieved at a unique point $t\in\mathbb{R}^{m}_{+}$ such that
$t_{i}u_{i}\in\mathcal{N}^{\alpha}$ for all $i=1,\cdots,m$; see [25, Eq.
(4.1)]. ∎

######  Remark 2.2.

Let $u\in H_{m}$ and $t\in\mathbb{R}^{m}_{+}$ such that
$t_{i}u_{i}\in\mathcal{N}^{\alpha}$ for all $i=1,\cdots,m$. Then

$t_{i}=\left(\frac{\int_{B^{N}}|\nabla
u_{i}|^{2}}{\int_{B^{N}}|x|^{\alpha}|u_{i}|^{p+1}}\right)^{\frac{1}{p-1}}\quad\forall
i=1,\cdots,m,$

and hence

$\varphi_{\alpha}(t_{i}u_{i})=\left(\frac{1}{2}-\frac{1}{p+1}\right)t_{i}^{2}\int_{B^{N}}|\nabla
u_{i}|^{2}=\left(\frac{1}{2}-\frac{1}{p+1}\right)\frac{\left(\int_{B^{N}}|\nabla
u_{i}|^{2}\right)^{\frac{p+1}{p-1}}}{\left(\int_{B^{N}}|x|^{\alpha}|u_{i}|^{p+1}\right)^{\frac{2}{p-1}}}.$

Consequently, by Proposition 2.1, we have the following characterization for
${C}_{\alpha,m}$:

${C}_{\alpha,m}=\left(\frac{1}{2}-\frac{1}{p+1}\right)\inf_{u\in
H_{m}}\sum_{i=1}^{m}\left[\frac{\left(\int_{B^{N}}|\nabla
u_{i}|^{2}\right)^{\frac{p+1}{p-1}}}{\left(\int_{B^{N}}|x|^{\alpha}|u_{i}|^{p+1}\right)^{\frac{2}{p-1}}}\right].$
(2.3)

######  Proposition 2.3.

There are constants $C_{1},C_{2}>0$, that depend only on $p$, $N$ and $m$,
such that

$C_{1}\left(\frac{\alpha+N}{N}\right)^{\frac{p+3}{p-1}}\leq{C}_{\alpha,m}\leq
C_{2}\left(\frac{\alpha+N}{N}\right)^{\frac{p+3}{p-1}},\quad\forall\alpha>\alpha_{p}.$

Moreover, the function $\alpha\mapsto C_{\alpha,m}$ is strictly increasing in
$(\alpha_{p},\infty)$.

###### Proof.

Here we use some arguments from the proof of [24, Theorem 4.1], where the case
of positive solutions are treated. Given $u\in H_{m}$, define the rescaled
function $v(t)=u(r)$, where $r=t^{\theta}$ and
$\theta=\frac{N}{\alpha+N}\in(0,1]$. Of course $v$ has $m$ nodal sets,
$v_{i}(t)=u_{i}(r)$ for each $i=1,\cdots,m$, and

$\displaystyle\int_{B^{N}}|x|^{\alpha}|u_{i}(x)|^{p+1}dx=\theta\int_{B^{N}}|v_{i}|^{p+1}dx,\qquad\int_{B^{N}}|\nabla
u_{i}(x)|^{2}dx=\theta^{-1}\int_{B^{N}}|\nabla
v_{i}(x)|^{2}|x|^{(2-N)(1-\theta)}dx.$

Then

$\frac{\left(\displaystyle\int_{B^{N}}|\nabla
u_{i}|^{2}\right)^{\frac{p+1}{p-1}}}{\left(\displaystyle\int_{B^{N}}|x|^{\alpha}|u_{i}|^{p+1}\right)^{\frac{2}{p-1}}}=\frac{1}{\theta^{\frac{p+3}{p-1}}}\frac{\left(\displaystyle\int_{B^{N}}|\nabla
v_{i}|^{2}|x|^{(2-N)(1-\theta)}\right)^{\frac{p+1}{p-1}}}{\left(\displaystyle\int_{B^{N}}|v_{i}|^{p+1}\right)^{\frac{2}{p-1}}}$

which, by (2.3), yields

${C}_{\alpha,m}=\frac{\left(\frac{1}{2}-\frac{1}{p+1}\right){R}_{\theta,m}}{\theta^{\frac{p+3}{p-1}}}=\left(\frac{1}{2}-\frac{1}{p+1}\right)\left(\frac{\alpha+N}{N}\right)^{\frac{p+3}{p-1}}{R}_{\theta,m}\,,$
(2.4)

where

${R}_{\rho,m}:=\inf_{v\in H_{m}}\sum_{i=1}^{m}\frac{\left(\int_{B^{N}}|\nabla
v_{i}|^{2}|x|^{(2-N)(1-\rho)}\right)^{\frac{p+1}{p-1}}}{\left(\int_{B^{N}}|v_{i}|^{p+1}\right)^{\frac{2}{p-1}}},\quad\rho\in[0,1].$

Since $\rho\mapsto{R}_{\rho,m}$ is non-increasing and
$0<{R}_{1,m}<{R}_{0,m}<\infty$, we infer that

$C_{1}\left(\frac{\alpha+N}{N}\right)^{\frac{p+3}{p-1}}\leq{C}_{\alpha,m}\leq
C_{2}\left(\frac{\alpha+N}{N}\right)^{\frac{p+3}{p-1}}\quad\forall\alpha>\alpha_{p},$

where $C_{1}=\left(\frac{1}{2}-\frac{1}{p+1}\right)\displaystyle\lim_{\rho\to
1^{-}}{R}_{\rho,m}$ and
$C_{2}=\left(\frac{1}{2}-\frac{1}{p+1}\right)\displaystyle\lim_{\rho\to
0^{+}}{R}_{\rho,m}$. The strict monotonicity of $\alpha\mapsto{C}_{\alpha,m}$
follows immediately from (2.4). ∎

##  3\. Estimates for the $L^{\infty}$-norms

In this section we present asymptotic estimates for $\|u_{\alpha}\|_{\infty}$.
We recall that, by [14, Lemma 5.2], $u_{\alpha}(0)=\displaystyle\max_{x\in
B^{N}}u_{\alpha}(x)=\|u_{\alpha}\|_{\infty}$. Moreover, by Proposition 2.3,
there exist constants $C_{1},C_{2}>0$, that do not depend on $\alpha$, such
that

$C_{1}\left(\frac{\alpha+N}{N}\right)^{\frac{p+3}{p-1}}\leq\int_{B^{N}}|\nabla
u_{\alpha}|^{2}=\int_{B^{N}}|x|^{\alpha}|u_{\alpha}|^{p+1}\leq
C_{2}\left(\frac{\alpha+N}{N}\right)^{\frac{p+3}{p-1}},\quad\forall\alpha>\alpha_{p}.$
(3.1)

We start with the bound from below.

######  Lemma 3.1.

There exists a constant $C>0$, that does not depend of $\alpha$, such that

$\|u_{\alpha}\|_{\infty}\geq
C\left(\frac{\alpha+N}{N}\right)^{\frac{2}{p-1}},\quad\forall\alpha>\alpha_{p}.$

###### Proof.

For each $\alpha>\alpha_{p}$, with $\theta=\frac{N}{\alpha+N}$, consider the
function

$\widetilde{u}_{\alpha}(y):=\theta^{\frac{2}{p-1}}u_{\alpha}\left(\theta^{\frac{1}{2-N}}y\right),\quad
y\in\Omega_{\alpha}:=B\left(0,\theta^{\frac{1}{N-2}}\right).$

Since
$\|\widetilde{u}_{\alpha}\|_{\infty}=\left(\frac{N}{\alpha+N}\right)^{\frac{2}{p-1}}\|u_{\alpha}\|_{\infty}$,
it suffices to show that $\|\widetilde{u}_{\alpha}\|_{\infty}\geq C$.
Performing the change of variables $x=\theta^{\frac{1}{2-N}}y$, we infer that

$\displaystyle\int_{\Omega_{\alpha}}|\nabla\widetilde{u}_{\alpha}(y)|^{2}dy=\left(\frac{N}{\alpha+N}\right)^{\frac{p+3}{p-1}}\int_{B^{N}}|\nabla
u_{\alpha}(x)|^{2}dx,$

whence, by (3.1),

$C_{1}\leq\int_{\Omega_{\alpha}}|\nabla\widetilde{u}_{\alpha}|^{2}dy\leq
C_{2},\quad\forall\alpha>\alpha_{p}.$ (3.2)

In addition, since $u_{\alpha}$ solves ($P_{\alpha}$),
$\widetilde{u}_{\alpha}$ satisfies

$-\Delta\widetilde{u}_{\alpha}(y)=\theta^{\frac{-2(N-1)}{N-2}}\left|\theta^{\frac{1}{2-N}}y\right|^{\alpha}|{\widetilde{u}_{\alpha}(y)}|^{p-1}\widetilde{u}_{\alpha}(y),\quad
y\in\Omega_{\alpha},\quad\widetilde{u}_{\alpha}=0\ \text{on}\
\partial\Omega_{\alpha}.$ (3.3)

Consequently, by Hölder’s inequality, (3.2) and (3.3),

$\displaystyle 0<C_{1}$
$\displaystyle\leq\int_{\Omega_{\alpha}}|\nabla\widetilde{u}_{\alpha}|^{2}dy=\int_{\Omega_{\alpha}}\theta^{\frac{-2(N-1)}{N-2}}\left|\theta^{\frac{1}{2-N}}y\right|^{\alpha}|{\widetilde{u}_{\alpha}}|^{p+1}dy$
(3.4)
$\displaystyle\leq\|{\widetilde{u}_{\alpha}}\|_{\infty}^{p-1}\int_{\Omega_{\alpha}}\theta^{\frac{-2(N-1)}{N-2}}\left|\theta^{\frac{1}{2-N}}y\right|^{\alpha}|{\widetilde{u}_{\alpha}}|^{2}dy$
$\displaystyle\leq\|{\widetilde{u}_{\alpha}}\|_{\infty}^{p-1}\left(\int_{\Omega_{\alpha}}\\!\\!\\!\\!\\!\theta^{\frac{-2(N-1)}{N-2}}\left|\theta^{\frac{1}{2-N}}y\right|^{\alpha}dy\right)^{\frac{p-1}{p+1}}\left(\int_{\Omega_{\alpha}}\\!\\!\\!\\!\\!\theta^{\frac{-2(N-1)}{N-2}}\left|\theta^{\frac{1}{2-N}}y\right|^{\alpha}|{\widetilde{u}_{\alpha}}|^{p+1}dy\right)^{\frac{2}{p+1}}$
$\displaystyle\leq(C_{2})^{\frac{2}{p+1}}\|{\widetilde{u}_{\alpha}}\|_{\infty}^{p-1}\left(\int_{\Omega_{\alpha}}\\!\\!\\!\\!\\!\theta^{\frac{-2(N-1)}{N-2}}\left|\theta^{\frac{1}{2-N}}y\right|^{\alpha}dy\right)^{\frac{p-1}{p+1}}.$

Note that, since $\theta=\frac{N}{\alpha+N}$,

$\displaystyle\int_{\Omega_{\alpha}}\theta^{\frac{-2(N-1)}{N-2}}\left|\theta^{\frac{1}{2-N}}y\right|^{\alpha}dy=\omega_{N-1}\theta^{\frac{2(N-1)+\alpha}{2-N}}\int_{0}^{\theta^{\frac{1}{N-2}}}t^{\alpha+N-1}dt=\frac{\omega_{N-1}}{N}.$
(3.5)

Therefore, from (3.4) and (3.5), there exists $C>0$, independent of $\alpha$,
such that

$\|{\widetilde{u}_{\alpha}}\|_{\infty}\geq C,\quad\forall\alpha>\alpha_{p},$

which proves the lemma. ∎

To prove the reverse estimate, for each $\alpha>\alpha_{p}$, consider

$v_{\alpha}(t):=\left(\frac{2}{\alpha+2}\right)^{\frac{2}{p-1}}u_{\alpha}\left(t^{\frac{2}{\alpha+2}}\right),\quad
t\in[0,1].$ (3.6)

Then by [9, Proposition 4.2], $v_{\alpha}$ satisfies

$\left\\{\begin{array}[]{l}-(t^{M_{\alpha}-1}v_{\alpha}^{\prime})^{\prime}=t^{M_{\alpha}-1}|v_{\alpha}|^{p-1}v_{\alpha}\quad\textrm{in}\
\ (0,1),\vspace{0.3 cm}\\\
v_{\alpha}(1)=v_{\alpha}^{\prime}(0)=0,\end{array}\right.$ (3.7)

where $M_{\alpha}:=\frac{2(\alpha+N)}{\alpha+2}\in(2,N)$. Then, integrating
from $0$ to $t$,

$v_{\alpha}^{\prime}(t)=-\frac{1}{t^{M_{\alpha}-1}}\int_{0}^{t}s^{M_{\alpha}-1}|v_{\alpha}(s)|^{p-1}v_{\alpha}(s)ds\quad\forall
t\in(0,1].$ (3.8)

Moreover, performing the change of variables $t\leftrightarrow r$, where
$r=t^{\frac{2}{\alpha+2}}$, we obtain

$\displaystyle\int_{0}^{1}|v_{\alpha}^{\prime}(t)|^{2}t^{M_{\alpha}-1}dt=\left(\frac{2}{\alpha+2}\right)^{\frac{p+3}{p-1}}\int_{0}^{1}|u_{\alpha}^{\prime}(r)|^{2}r^{N-1}dr.$
(3.9)

######  Lemma 3.2.

There exist $\alpha_{0}>\alpha_{p}$ and $C>0$, that do not depend on $\alpha$,
such that

$\|u_{\alpha}\|_{\infty}\leq C\left(\frac{\alpha+2}{2}\right)^{\frac{2}{p-1}}\
\ \forall\,\alpha\geq\alpha_{0}.$

###### Proof.

By (3.6), it is enough to show that

$v_{\alpha}(0)=\|v_{\alpha}\|_{\infty}\leq C,\ \
\forall\,\alpha\geq\alpha_{0}.$

From (3.1) and (3.9), there exists a constant $C_{0}>0$ independent of
$\alpha$ such that

$\int_{0}^{1}|v_{\alpha}^{\prime}(t)|^{2}t^{M_{\alpha}-1}dt\leq
C_{0}\quad\forall\alpha>\alpha_{p}.$ (3.10)

Fix $M>2$, with $p<\frac{M}{M-2}$, and take $\alpha_{0}>0$ so that
$M_{\alpha}\leq M$ for $\alpha\geq\alpha_{0}$. By contradiction, suppose that
$\|v_{\alpha}\|_{\infty}$ is not bounded in $[\alpha_{0},\infty)$. Then there
exists a sequence $\alpha_{n}\geq\alpha_{0}$ such that
$\|v_{\alpha_{n}}\|_{\infty}\to\infty$ as $n\to\infty$. Set

$\overline{v}_{\alpha}(s)=\frac{1}{\|v_{\alpha}\|_{\infty}}v_{\alpha}\left(\|v_{\alpha}\|_{\infty}^{1-p}s\right),\quad\text{for}\
s\in\left[0,\|v_{\alpha}\|_{\infty}^{p-1}\right],\quad\text{and}\ \
t=\|v_{\alpha}\|_{\infty}^{1-p}s.$

Then, since $M\geq M_{\alpha_{n}}$, from (3.10) and $(M-2)p-M<0$, we infer
that

$\displaystyle\int_{0}^{\|v_{\alpha_{n}}\|_{\infty}^{p-1}}|\overline{v}_{\alpha_{n}}^{\prime}(s)|^{2}s^{M-1}ds$
$\displaystyle=\|v_{\alpha_{n}}\|_{\infty}^{(M-2)p-M}\int_{0}^{1}|v_{\alpha_{n}}^{\prime}(t)|^{2}t^{M-1}dt$
$\displaystyle\leq\|v_{\alpha_{n}}\|_{\infty}^{(M-2)p-M}\int_{0}^{1}|v_{\alpha_{n}}^{\prime}(t)|^{2}t^{M_{\alpha_{n}}-1}dt\leq
C_{0}\|v_{\alpha_{n}}\|_{\infty}^{(M-2)p-M}\to 0.$

Hence $\overline{v}_{\alpha_{n}}\to 0\ \text{in}\ \mathcal{D}^{1,2}_{M},$
where

$\mathcal{D}^{1,2}_{M}:=\left\\{w\in
L^{2^{*}_{M}}_{M}(0,\infty):\quad\begin{aligned} &\text{$w$ has first order
weak derivative and}\\\
&\|w\|_{\mathcal{D}^{1,2}_{M}}^{2}:=\int_{0}^{\infty}|w^{\prime}(t)|^{2}t^{M-1}dt<\infty\end{aligned}\right\\}$

and

$L^{2^{*}_{M}}_{M}(0,\infty):=\left\\{v:[0,\infty)\to\mathbb{R}\
\text{measurable}:\int_{0}^{\infty}|v(s)|^{\frac{2M}{M-2}}s^{M-1}ds<\infty\right\\}.$

Since, by Sobolev’s inequality, the inclusion $\mathcal{D}^{1,2}_{M}\subset
L^{2^{*}_{M}}_{M}(0,\infty)$ is continuous, up to a subsequence,
$\overline{v}_{\alpha_{n}}\to 0$ a.e. in $[0,\infty)$.

On the other hand,

$\overline{v}_{\alpha_{n}}(0)=\|\overline{v}_{\alpha_{n}}\|_{\infty}=1,$

and thus, the sequence of functions $(\overline{v}_{\alpha_{n}})$ is uniformly
bounded. Moreover, by (3.8),

$|\overline{v}_{\alpha_{n}}^{\prime}(s)|=\|v_{\alpha_{n}}\|_{\infty}^{-p}|v_{\alpha_{n}}^{\prime}\left(\|v_{\alpha_{n}}\|_{\infty}^{1-p}s\right)|\leq\|v_{\alpha_{n}}\|_{\infty}^{-p}\|v_{\alpha_{n}}\|_{\infty}^{p}=1.$

In particular, the sequence of functions $(\overline{v}_{\alpha_{n}})$ is
equicontinuous. Consequently, by Arzelà-Ascoli theorem, up to a subsequence,
$\overline{v}_{\alpha_{n}}\to\overline{v}$ uniformly in $[0,1]$ with
$\|\overline{v}\|_{\infty}=1$, which leads us to a contradiction. ∎

######  Remark 3.3.

Notice that, unlike Lemma 3.1, the conclusion of the previous lemma is not
true for all $\alpha>\alpha_{p}$. Consider, for example, $m=1$ ($u_{\alpha}$
is positive) and $p=\frac{N+2}{N-2}$ ($\alpha_{p}=0$). In this case,
$\|u_{\alpha}\|_{\infty}\to\infty$ as $\alpha\to 0$. Indeed, suppose that
$\|u_{\alpha}\|_{\infty}$ is uniformly bounded as $\alpha\to 0$. Thus, since
$u_{\alpha}$ solves

$-\Delta u=|x|^{\alpha}u^{\frac{N+2}{N-2}},\ u>0\ \text{in}\ B^{N},\quad u=0\
\text{on}\ \partial B^{N},$

we have that as $\alpha\to 0$, $u_{\alpha}$ converges uniformly to a
nonnegative solution $u$ of the limit problem

$-\Delta u=u^{\frac{N+2}{N-2}}\ \text{in}\ B^{N},\quad u=0\ \text{on}\
\partial B^{N}.$

This implies that $u=0$, which is a contradiction, because
$\|u_{\alpha}\|_{\infty}\geq C$ by Lemma 3.1.

##  4\. Estimates for the local extrema and proof of Theorem 1.6 (iii)

If $m=1$, then $x=0$ is the unique local extremum of $u_{\alpha}$, and Lemmas
3.1 and 3.2 give a sharp estimate for $u_{\alpha}(0)$. Next, if $m\geq 2$,
denote by $r_{1,\alpha}<\cdots<r_{m,\alpha}=1$ the zeros of $u_{\alpha}$ in
$(0,1]$ and set

$\mathcal{M}_{1,\alpha}:=\max\\{u_{\alpha}(r):0\leq r\leq
r_{1,\alpha}\\},\quad\mathcal{M}_{i+1,\alpha}:=\max\\{|u_{\alpha}(r)|:r_{i,\alpha}\leq
r\leq r_{i+1,\alpha}\\}$

for $i=1,\cdots,m-1$. Similarly, denote by
$t_{1,\alpha}<\cdots<t_{m,\alpha}=1$ the zeros in $(0,1]$ of the function
$v_{\alpha}$ defined in (3.6). Then

$t_{i,\alpha}=(r_{i,\alpha})^{\frac{\alpha+2}{2}},\quad\text{for each}\
i=1,\cdots,m.$ (4.1)

Moreover, define

$\mathcal{N}_{1,\alpha}:=\max\\{v_{\alpha}(t):0\leq t\leq
t_{1,\alpha}\\},\quad\mathcal{N}_{i+1,\alpha}:=\max\\{|v_{\alpha}(t)|:t_{i,\alpha}\leq
t\leq t_{i+1,\alpha}\\},$

and observe that

$\mathcal{N}_{i,\alpha}=\left(\frac{2}{\alpha+2}\right)^{\frac{2}{p-1}}\mathcal{M}_{i,\alpha},\quad\text{for
each}\ i=1,\cdots,m.$ (4.2)

######  Lemma 4.1.

There exists $\delta\in(0,1)$ and $\alpha_{0}>\alpha_{p}$ such that

$t_{1,\alpha}\geq\delta\quad\forall\alpha\geq\alpha_{0}.$

In particular, by (4.1), $r_{i,\alpha}\to 1$ as $\alpha\to\infty$, for each
$i=1,\cdots,m$.

###### Proof.

Let $\alpha_{0}>\alpha_{p}$ be as in Lemma 3.2. By (3.6) and Lemmas 3.1 and
3.2, there exist constants $C_{1},C_{2}>0$ such that

$C_{1}\leq\|v_{\alpha}\|_{\infty}\leq C_{2},\quad\text{for all
$\alpha\geq\alpha_{0}$}.$ (4.3)

Moreover, by (3.8),

$\|{v_{\alpha}}^{\prime}\|_{\infty}\leq(C_{2})^{p},\quad\text{for all
$\alpha\geq\alpha_{0}$}.$ (4.4)

Suppose, by contradiction, that there exists a sequence
$\alpha_{n}\geq\alpha_{0}$ such that $t_{1,\alpha_{n}}\to 0$ as $n\to\infty$.
By (4.3) and (4.4), we may use Arzelà-Ascoli theorem to conclude that, up to a
subsequence, $v_{\alpha_{n}}\to v$ uniformly in $[0,1]$ as $n\to\infty$. Note
that

$v(0)=v(0)-v_{\alpha_{n}}(t_{1,\alpha_{n}})=[v(0)-v(t_{1,\alpha_{n}})]+[v(t_{1,\alpha_{n}})-v_{\alpha_{n}}(t_{1,\alpha_{n}})]=o(1),$

that is, $v(0)=0$. On the other hand

$0<C_{1}\leq\lim_{n\to\infty}\|v_{\alpha_{n}}\|_{\infty}=\lim_{n\to\infty}v_{\alpha_{n}}(0)=v(0),$

which is a contradiction. ∎

######  Remark 4.2.

Observe that the previous lemma guarantees asymptotic concentration, as
$\alpha\to\infty$, of the zeros of the radial solutions $u_{\alpha}$ of
($P_{\alpha}$), namely $r_{i,\alpha}\to 1$ for each $i=1,\cdots,m$. We improve
this result in Section 5, proving Theorem 1.6, where we show the exact rate at
which this concentration occurs.

Next, we obtain some asymptotic estimates for the local extrema
$\mathcal{M}_{i,\alpha}$ of $u_{\alpha}$, as $\alpha\to\infty$.

###### Proof of Theorem 1.6. Part (iii).

By (4.2), it is enough to prove that

$C_{1}\leq\mathcal{N}_{i,\alpha}\leq C_{2},\ \ \forall\ \ i=1,\cdots,m,\ \
\text{and}\ \ \alpha\geq\alpha_{0}.$

By [4, Lemma 3.1],
$\mathcal{N}_{1,\alpha}>\mathcal{N}_{2,\alpha}>\cdots>\mathcal{N}_{m,\alpha}$
for each $\alpha>\alpha_{p}$. Therefore, we may take $C_{2}$ as the constant
$C$ from Lemma 3.2.

Next for simplicity, we denote by $t_{\alpha}=t_{m-1,\alpha}$ the largest zero
of $v_{\alpha}$ in (0,1) so that
$\mathcal{N}_{m,\alpha}=\displaystyle\max_{t_{\alpha}<t<1}|v_{\alpha}(t)|$.
Multiplying (3.7) by $v_{\alpha}$ and integrating by parts from $t_{\alpha}$
to $1$, we get

$\displaystyle\int_{t_{\alpha}}^{1}|v_{\alpha}(t)|^{p+1}t^{M_{\alpha}-1}dt=\int_{t_{\alpha}}^{1}|v_{\alpha}^{\prime}(t)|^{2}t^{M_{\alpha}-1}dt,$

since $v_{\alpha}(t_{\alpha})=v_{\alpha}(1)=0$. Moreover, for every
$t\in(t_{\alpha},1)$,

$\displaystyle|v_{\alpha}(t)|$
$\displaystyle=\left|\int_{t_{\alpha}}^{t}v_{\alpha}^{\prime}(s)ds\right|\leq\int_{t_{\alpha}}^{1}\left(s^{\frac{1-M_{\alpha}}{2}}\right)\left(|v_{\alpha}^{\prime}(s)|s^{\frac{M_{\alpha}-1}{2}}\right)ds$
$\displaystyle\leq\left(\int_{t_{\alpha}}^{1}s^{1-M_{\alpha}}ds\right)^{1/2}\left(\int_{t_{\alpha}}^{1}|v_{\alpha}^{\prime}(s)|^{2}s^{M_{\alpha}-1}ds\right)^{1/2}.$

By Lemma (4.1), there is $\delta>0$ such that $t_{\alpha}\geq\delta$ and hence

$\left(\int_{t_{\alpha}}^{1}s^{1-M_{\alpha}}ds\right)^{1/2}\leq\left(\int_{\delta}^{1}s^{1-N}ds\right)^{1/2}=\left(\frac{\delta^{2-N}-1}{N-2}\right)^{1/2}=:C.$

Consequently,

$|v_{\alpha}(t)|\leq
C\left(\int_{t_{\alpha}}^{1}|v_{\alpha}^{\prime}(s)|^{2}s^{M_{\alpha}-1}ds\right)^{1/2}=C\left(\int_{t_{\alpha}}^{1}|v_{\alpha}(s)|^{p+1}s^{M_{\alpha}-1}ds\right)^{1/2},\quad\forall
t\in(t_{\alpha},1),$

which implies, by definition of $\mathcal{N}_{m,\alpha}$, that

$\mathcal{N}_{m,\alpha}\leq
C\left(\int_{t_{\alpha}}^{1}|v_{\alpha}(s)|^{p+1}s^{M_{\alpha}-1}ds\right)^{1/2}\leq
C(\mathcal{N}_{m,\alpha})^{(p+1)/2},$

that is,

$\mathcal{N}_{m,\alpha}\geq\left(\frac{1}{C}\right)^{2/(p-1)}>0.$

Taking $C_{1}=C^{2/(1-p)}$, we obtain
$\mathcal{N}_{1,\alpha}>\mathcal{N}_{2,\alpha}>\cdots>\mathcal{N}_{m,\alpha}\geq
C_{1}$, which concludes the proof. ∎

##  5\. Proofs of Theorems 1.1, 1.6 and 1.8

###### Proof of Theorem 1.1.

From Lemma 3.2, $\|v_{\alpha}\|_{\infty}\leq C$ for all $\alpha>\alpha_{0}$.
By (3.8), this implies that $|v^{\prime}_{\alpha}(t)|\leq C^{p}|t|$ for all
$t\in(0,1]$, and hence $\|v^{\prime}_{\alpha}\|_{\infty}\leq C^{p}$. Moreover,
since $v_{\alpha}$ satisfies (3.7), for all $t\in(0,1]$ and
$\alpha>\alpha_{0}$,

$|v_{\alpha}^{\prime\prime}(t)|=\left|(M_{\alpha}-1)\frac{v_{\alpha}^{\prime}(t)}{t}+|v_{\alpha}(t)|^{p-1}v_{\alpha}(t)\right|\leq(N-1)C^{p}+C^{p}=NC^{p},$

that is, $\|v_{\alpha}^{\prime\prime}\|_{\infty}\leq NC^{p}$.

Let $(\alpha_{n})$ be any sequence such that $\alpha_{n}\to\infty$ as
$n\to\infty$. Then, by Arzelà-Ascoli theorem, up to a subsequence,
$v_{\alpha_{n}}\to v$ and $v_{\alpha_{n}}^{\prime}\to z$ uniformly in $[0,1]$.
Then $v$ is differentiable and $v^{\prime}=z$. We claim that $v\in
H^{1}_{0,rad}(B^{2})$ is a weak radial solution of ($L$). Indeed, let $\psi\in
C^{1}([0,1])$ be such that $\psi(1)=0$. Then, multiplying (3.7) by $\psi$ and
integrating by parts,

$\int_{0}^{1}v_{\alpha_{n}}^{\prime}(t)\psi^{\prime}(t)t^{M_{\alpha_{n}}-1}dt=\int_{0}^{1}|v_{\alpha_{n}}(t)|^{p-1}v_{\alpha_{n}}(t)\psi(t)t^{M_{\alpha_{n}}-1}dt,\quad\forall
n\in\mathbb{N}.$

Hence, by uniform convergence,

$\int_{0}^{1}v^{\prime}(t)\psi^{\prime}(t)\,t\,dt=\int_{0}^{1}|v(t)|^{p-1}v(t)\psi(t)\,t\,dt,$

since $M_{\alpha_{n}}\to 2$. This implies that $v$ is a weak radial solution
of ($L$) and, by standard elliptic regularity, $v\in C^{2}([0,1])$ and
satisfies

$v^{\prime\prime}+\frac{v^{\prime}}{t}+|v|^{p-1}v=0\quad\text{in}\ (0,1),\quad
v^{\prime}(0)=v(1)=0.$

Given $\varepsilon\in(0,1)$, for all $t\in[\varepsilon,1]$, we may write

$\displaystyle|v_{\alpha}^{\prime\prime}(t)-v^{\prime\prime}(t)|$
$\displaystyle\leq\frac{|v^{\prime}(t)-(M_{\alpha}-1)v_{\alpha}^{\prime}(t)|}{t}+\left||v(t)|^{p-1}v(t)-|v_{\alpha}(t)|^{p-1}v_{\alpha}(t)\right|$
$\displaystyle\leq\frac{|v^{\prime}(t)-(M_{\alpha}-1)v_{\alpha}^{\prime}(t)|}{\varepsilon}+\left||v(t)|^{p-1}v(t)-|v_{\alpha}(t)|^{p-1}v_{\alpha}(t)\right|,$

which shows that $v^{\prime\prime}_{\alpha_{n}}\to v^{\prime\prime}$ in
$C([\varepsilon,1])$. Putting all these convergences together, we infer that
$v_{\alpha_{n}}\to v$ in $C^{1}([0,1])\cap C^{2}([\varepsilon,1])$.

To conclude the proof, we need to check that $v=w$. Since $v$ and $w$ are
radial solutions of ($L$) and $w$ has precisely $m$ nodal sets, it is enough
to show that $v$ also has $m$ nodal sets. Up to a subsequence, we can assume
that $t_{i,\alpha_{n}}\to t_{i}\in[0,1]$ for each $i=1,\cdots,m$. Of course
$0<\delta\leq t_{1}\leq t_{2}\leq\cdots\leq t_{m}=1$, where $\delta$ is given
by Lemma 4.1. We claim that $t_{i}<t_{i+1}$ for all $i=1,\cdots,m-1$. Indeed,
by contradiction, suppose that $t_{i}=t_{i+1}$ for some $i$. Let
$s_{\alpha_{n}}\in(t_{i,\alpha_{n}},t_{i+1,\alpha_{n}})$ so that
$|v_{\alpha_{n}}(s_{\alpha_{n}})|=\mathcal{N}_{i+1,\alpha_{n}}$. Then, by
Theorem 1.6 (iii),

$\left|\frac{v_{\alpha_{n}}(s_{\alpha_{n}})-v_{\alpha_{n}}(t_{i,\alpha_{n}})}{s_{\alpha_{n}}-t_{i,\alpha_{n}}}\right|=\frac{|v_{\alpha_{n}}(s_{\alpha_{n}})|}{s_{\alpha_{n}}-t_{i,\alpha_{n}}}\geq\frac{C_{1}}{s_{\alpha_{n}}-t_{i,\alpha_{n}}}\to\infty,\quad
n\to\infty,$

which contradicts $\|v^{\prime}_{\alpha}\|_{\infty}\leq C^{p}$. Finally, since
$|v_{\alpha_{n}}|>0$ in $(0,t_{1,\alpha})$ and
$(t_{i,\alpha_{n}},t_{i+1,\alpha_{n}})$, for all $i=1,\ldots,m-1$, we infer
that $|v|\geq 0$ in $(0,t_{1})$ and $(t_{i},t_{i+1})$ for all
$i=1,\ldots,m-1$. Then, by the strong maximum principle and Theorem 1.6 (iii),
$|v|>0$ in each of the intervals $(0,t_{1})$ and $(t_{i},t_{i+1})$ for all
$i=1,\ldots,m-1$, and hence $v$ has $m$ nodal sets. ∎

######  Remark 5.1.

Notice that, for all ${\alpha}^{*}>\alpha_{p}$, using the same arguments, it
is possible to show that the functions $v_{\alpha}$ converge in
$C^{1}([0,1])\cap C^{2}([\varepsilon,1])$ to the function $v_{{\alpha}^{*}}$,
as $\alpha\to{\alpha}^{*}$. This can be reformulated as follows. First, take
into account that $v_{\alpha}$ is the unique solution with $m$ nodal sets of
the equation

$\left\\{\begin{array}[]{l}-v^{\prime\prime}-\frac{M-1}{t}v^{\prime}=|v|^{p-1}v\quad\textrm{in}\
\ (0,1),\vspace{0.3 cm}\\\ v(0)>0,\ v^{\prime}(0)=v(1)=0,\end{array}\right.$
($Q_{M}$)

with $M=M_{\alpha}=\frac{2(\alpha+N)}{\alpha+2}\in(2,N)$ (see [9, Proposition
4.2]). Then for all $M_{0}>2$ such that
$p+1<2^{*}_{M_{0}}:=\frac{2M_{0}}{M_{0}-2}$, that is,
$2<M_{0}<\frac{2(p+1)}{p-1}$, the solution $v_{M}$ of ($Q_{M}$) with $m$ nodal
sets converges in $C^{1}([0,1])\cap C^{2}([\varepsilon,1])$ to the solution
with $m$ nodal sets of $(P_{M_{0}})$, as $M\to M_{0}$.

###### Proof of Theorem 1.6.

Part (i). By Theorem 1.1, $t_{i,\alpha}\to t_{i}$, for each $i=1,\cdots,m-1$,
where $t_{i,\alpha}$ are the zeros of $v_{\alpha}$ defined in (1.1). Since
$t_{i,\alpha}=r_{i,\alpha}^{\frac{\alpha+2}{2}}$,

$\frac{\alpha+2}{2}(1-r_{i,\alpha})=\frac{\alpha+2}{2}\left(1-t_{i,\alpha}^{\frac{2}{2+\alpha}}\right)=\frac{\alpha+2}{2}\left(1-t_{i}^{\frac{2}{2+\alpha}}+t_{i}^{\frac{2}{2+\alpha}}-t_{i,\alpha}^{\frac{2}{2+\alpha}}\right).$

Note that, by the mean value theorem, there is $c_{i,\alpha}$ between $t_{i}$
and $t_{i,\alpha}$ such that

$\frac{\alpha+2}{2}\left(t_{i}^{\frac{2}{\alpha+2}}-t_{i,\alpha}^{\frac{2}{\alpha+2}}\right)=\frac{\alpha+2}{2}\left(\frac{2}{\alpha+2}c_{i,\alpha}^{\frac{-\alpha}{\alpha+2}}(t_{i}-t_{i,\alpha})\right)=c_{i,\alpha}^{\frac{-\alpha}{\alpha+2}}(t_{i}-t_{i,\alpha})\to
0,$

because $t_{i,\alpha}\to t_{i}$ and
$c_{i,\alpha}^{\frac{-\alpha}{\alpha+2}}\to 1/t_{i}$. Therefore, as
$\alpha\to\infty$,

$\frac{\alpha+2}{2}(1-r_{i,\alpha})=\frac{\alpha+2}{2}\left(1-t_{i}^{\frac{2}{\alpha+2}}\right)+o(1)\to-\log(t_{i}),$

and hence

$\alpha(1-r_{i,\alpha})\to-2\log(t_{i}),\quad\text{as}\ \alpha\to\infty.$

Part (ii).  First we prove that for each $x\in B^{N}$ fixed,

$\lim_{\alpha\to\infty}\left(\frac{2}{\alpha+2}\right)^{\frac{2}{p-1}}u_{\alpha}(x)=w(0)=\|w\|_{\infty}.$
(5.1)

Indeed, fixed $x\in B^{N}$, set $r=|x|\in[0,1)$. Since $(v_{\alpha})$
converges uniformly to $w$,

$\left(\frac{2}{\alpha+2}\right)^{\frac{2}{p-1}}u_{\alpha}(r)=v_{\alpha}(r^{\frac{\alpha+2}{2}})=w(r^{\frac{\alpha+2}{2}})+\left[v_{\alpha}(r^{\frac{\alpha+2}{2}})-w(r^{\frac{\alpha+2}{2}})\right]=w(0)+o(1),$

as $\alpha\to\infty$, which proves (5.1).

Next, we prove the $L^{\infty}_{loc}$ blow-up of $u_{\alpha}$. From the
equation

$-(r^{N-1}u_{\alpha}^{\prime})^{\prime}=r^{N-1}|u_{\alpha}|^{p-1}u_{\alpha},\quad
u_{\alpha}(0)>0,\quad u_{\alpha}^{\prime}(0)=u_{\alpha}(1)=0,$

we infer that

$u_{\alpha}^{\prime}(r)=-\frac{1}{r^{N-1}}\int_{0}^{r}s^{N-1}|u_{\alpha}(s)|^{p-1}u_{\alpha}(s)ds,\quad\forall
r\in(0,1).$

By (i), $r_{1,\alpha}\to 1$ as $\alpha\to\infty$. Hence, given $0<R_{0}<1$,
there is $\alpha_{1}>\alpha_{p}$ such that $r_{1,\alpha}>R_{0}$ for all
$\alpha>\alpha_{1}$. Consequently, $u_{\alpha}(r)>0$ for all $r\in[0,R_{0}]$
and $\alpha>\alpha_{1}$. Thus $u_{\alpha}^{\prime}(r)<0$ for $r\in[0,R_{0}]$
and $\alpha>\alpha_{1}$. Set
$W_{\alpha}(r):=\left(\frac{2}{\alpha+2}\right)^{\frac{2}{p-1}}u_{\alpha}(r)$.
Then, given $\varepsilon>0$, by (5.1), there is $\alpha_{2}>\alpha_{1}$ such
that

$|W_{\alpha}(0)-W_{\alpha}(R_{0})|<\varepsilon/2,\quad|W_{\alpha}(R_{0})-w(0)|<\varepsilon/2\quad\forall\alpha>\alpha_{2}.$

Theorefore, the monotonicity of $W_{\alpha}$ in $[0,R_{0}]$ for
$\alpha>\alpha_{2}$ implies that for any $r\in[0,R_{0}]$

$\displaystyle|W_{\alpha}(r)-w(0)|$
$\displaystyle\leq|W_{\alpha}(r)-W_{\alpha}(R_{0})|+|W_{\alpha}(R_{0})-w(0)|=W_{\alpha}(r)-W_{\alpha}(R_{0})+|W_{\alpha}(R_{0})-w(0)|$
$\displaystyle\leq
W_{\alpha}(0)-W_{\alpha}(R_{0})+|W_{\alpha}(R_{0})-w(0)|<\varepsilon,\quad\forall\alpha>\alpha_{2}.$

From this we get the uniform convergence, since $\alpha_{2}$ does not depend
on $r$. ∎

###### Proof of Theorem 1.8.

By compact embedding, $S_{p}$ is achieved at a positive solution $w$ of ($L$),
which, by [12], is radially symmetric. Moreover, by (3.9) and Theorem 1.1, as
$\alpha\to\infty$,

$\displaystyle\left(\frac{2}{\alpha+2}\right)^{\frac{p+3}{p-1}}\int_{B^{N}}|\nabla
u_{\alpha}|^{2}$
$\displaystyle=\omega_{N-1}\int_{0}^{1}|v_{\alpha}^{\prime}(t)|^{2}t^{M_{\alpha}-1}dt$
$\displaystyle\to\omega_{N-1}\int_{0}^{1}|w^{\prime}(t)|^{2}\,t\,dt=\frac{\omega_{N-1}}{2\pi}\int_{B^{2}}|\nabla
w|^{2}.$

Then, since

$\int_{B^{N}}|\nabla
u_{\alpha}|^{2}=(S_{\alpha,p}^{R})^{\frac{p+1}{p-1}}\quad\text{and}\quad\int_{B^{2}}|\nabla
w|^{2}=(S_{p})^{\frac{p+1}{p-1}},$

we infer that

$\left(\frac{2}{\alpha+2}\right)^{\frac{p+3}{p+1}}S_{\alpha,p}^{R}\to\left(\frac{\omega_{N-1}}{2\pi}\right)^{\frac{p-1}{p+1}}S_{p},\quad\text{as}\
\ \alpha\to\infty.\qed$

##  6\. Asymptotic distribution for the spectrum of the linearized operators
and proofs of Theorems 1.3 and 1.5

Here we analyze the spectrum of some linear operators. More precisely, we
derive asymptotic expansions of the negative eigenvalues of the linearized
operators associated to the Hénon equation ($P_{\alpha}$) at the radial
solution $u_{\alpha}$, with $m$ nodal sets, as $\alpha\to\infty$. For this,
consider the linearized operators $L^{\alpha}:H^{2}(B^{N})\cap
H^{1}_{0}(B^{N})\to L^{2}(B^{N})$ given by

$\varphi\mapsto
L^{\alpha}\varphi:=-\Delta\varphi-p|x|^{\alpha}|u_{\alpha}|^{p-1}\varphi,\quad\alpha>\alpha_{p}.$

In order to obtain a more precise description of the distribution of the
eigenvalues of $L^{\alpha}$, as $\alpha\to\infty$, we consider the singular
eigenvalue problem

$L^{\alpha}\varphi={\widehat{\Lambda}}\frac{\varphi}{{|x|^{2}}}.$ (6.1)

Since $N\geq 3$, we recall that, due to the Hardy inequality, (6.1) is well
defined in $H^{1}_{0}(B^{N})$. Moreover, $\widehat{\Lambda}$ is an eigenvalue
for (6.1) if there exists $0\neq\varphi\in H^{1}_{0}(B^{N})$ such that

$\int_{B^{N}}\nabla\varphi\nabla\phi-p|x|^{\alpha}|u_{\alpha}|^{p-1}\varphi\phi\,dx=\widehat{\Lambda}\int_{B^{N}}\frac{\varphi\phi}{|x|^{2}}\,dx,\quad\forall\phi\in
H^{1}_{0}(B^{N}).$

In addition, from [3, Proposition 4.1], it is known that each of these
negative eigenvalues is given (and vice versa) by the following decomposition

$\widehat{\Lambda}=\widehat{\Lambda}^{rad}+j(N-2+j),$ (6.2)

where $\widehat{\Lambda}^{rad}$ is a negative radial eigenvalue of (6.1) and
$j$ is some nonnegative integer. So, the negative eigenvalues of (6.1) can be
given in terms of its negative radial eigenvalues. For this reason, we study
the asymptotic distribution of negative radial eigenvalues for (6.1).

Since $u_{\alpha}$ has $m$ nodal sets, (6.1) admits precisely $m$ negative
radial eigenvalues; see [4, Theorem 1.3]. Denote these eigenvalues by
${\widehat{\Lambda}}_{1,\alpha}<\cdots<{\widehat{\Lambda}}_{m,\alpha}$, and by
$\varphi_{i,\alpha}$ their respective eigenfunctions.

For each $M\geq 2$, set

$H^{1}_{0,M}:=\left\\{w:[0,1]\to\mathbb{R}\
\text{measurable}:\quad\begin{aligned} &\text{$w$ has first order weak
derivative, $w(1)=0$ and}\\\
&\int_{0}^{1}|w^{\prime}(t)|^{2}t^{M-1}dt+\int_{0}^{1}w^{2}(t)t^{M-3}dt<\infty\end{aligned}\right\\},$

which is a Hilbert space with inner product

$(z,v)\mapsto\int_{0}^{1}z^{\prime}(t)v^{\prime}(t)t^{M-1}dt+\int_{0}^{1}z(t)v(t)t^{M-3}dt.$

Observe, in particular, that we may rewrite $H^{1}_{0,2}$ as

$H^{1}_{0,2}:=\left\\{u\in
H^{1}_{0,rad}(B^{2});\int_{B^{2}}\frac{u^{2}}{|y|^{2}}<\infty\right\\}.$

Now let $w$ be a radial solution of ($L$) with $m$ nodal sets, and consider
the singular eigenvalue problem associated to ($L$) at $w$, namely

$-\Delta\psi-p|w|^{p-1}\psi=\lambda\frac{\psi}{|y|^{2}}\ \ \text{in}\ \
B^{2}\backslash\\{0\\},\quad\psi\in H^{1}_{0,2},$

which also has $m$ negative radial eigenvalues, say
$\lambda_{1}<\cdots<\lambda_{m}$.

######  Lemma 6.1.

$\lim_{\alpha\to\infty}\left(\frac{2}{\alpha+2}\right)^{2}\widehat{\Lambda}_{i,\alpha}=\lambda_{i},\quad\forall\,i=1,\cdots,m.$

###### Proof.

As before, we perform the change of variables $t=r^{\frac{\alpha+2}{2}}$,
$r=|x|$, and write

$v_{\alpha}(t)=\left(\frac{2}{\alpha+2}\right)^{\frac{2}{p-1}}u_{\alpha}(r),\quad\psi_{i,\alpha}(t)=\varphi_{i,\alpha}(r),\quad
t\in(0,1).$

Since $\varphi_{i,\alpha}$ is radial and solves (6.1) with
$\widehat{\Lambda}={\widehat{\Lambda}}_{i,\alpha}$, using [9, Proposition
4.2], $\psi_{i,\alpha}$ satisfies

$-\psi_{i,\alpha}^{\prime\prime}-\frac{M_{\alpha}-1}{t}\psi_{i,\alpha}^{\prime}-p|v_{\alpha}|^{p-1}\psi_{i,\alpha}=\lambda_{i,\alpha}\frac{\psi_{i,\alpha}}{t^{2}},\quad
t\in(0,1),\quad\psi_{i,\alpha}(1)=0,$ (6.3)

with $M_{\alpha}=\frac{2(\alpha+N)}{\alpha+2}$ and
$\lambda_{i,\alpha}=\left(\frac{2}{\alpha+2}\right)^{2}\widehat{\Lambda}_{i,\alpha}$.
Recall that, see [3, Eq. (3.24)], each eigenvalue $\lambda_{i,\alpha}$ is
characterized by

$\lambda_{i,\alpha}=\min_{\overset{Z\subset H^{1}_{0,M_{\alpha}}}{\dim
Z=i}}\max_{\overset{z\in Z}{z\neq
0}}\frac{\int_{0}^{1}[|z^{\prime}(t)|^{2}-p|v_{\alpha}|^{p-1}z^{2}(t)]t^{M_{\alpha}-1}dt}{\int_{0}^{1}z^{2}(t)t^{M_{\alpha}-3}dt}.$

and this min-max problem is attained at the subspace
$Z_{i}=span\\{\psi_{1,\alpha},\ldots,\psi_{i,\alpha}\\}$, at
$z_{i}=\psi_{i,\alpha}$. By [3, Proposition 3.8], the eigenfunctions
$\psi_{k,\alpha}$ lie on $H^{1}_{0,2}\subset H^{1}_{0,M_{\alpha}}$, for all
$k\in\\{1,\ldots,m\\}$ and $\alpha>\alpha_{p}$. Therefore,

$\lambda_{i,\alpha}=\min_{\overset{Z\subset H^{1}_{0,2}}{\dim
Z=i}}\max_{\overset{z\in Z}{z\neq
0}}\frac{\int_{0}^{1}[|z^{\prime}(t)|^{2}-p|v_{\alpha}|^{p-1}z^{2}(t)]t^{M_{\alpha}-1}dt}{\int_{0}^{1}z^{2}(t)t^{M_{\alpha}-3}dt},\quad\forall
i=1,\cdots,m.$ (6.4)

By Theorem 1.1, $|v_{\alpha}|^{p-1}\to|w|^{p-1}$ uniformly in $[0,1]$, as
$\alpha\to\infty$. Similarly to [17, Lemma 3.5], since $M_{\alpha}\to 2$ as
$\alpha\to\infty$, this implies that

$\lambda_{i,\alpha}\to\min_{\overset{Z\subset H^{1}_{0,2}}{\dim
Z=i}}\max_{\overset{z\in Z}{z\neq
0}}\frac{\int_{0}^{1}[|z^{\prime}(t)|^{2}-p|w|^{p-1}z^{2}(t)]\,t\,dt}{\int_{0}^{1}z^{2}(t)t^{-1}dt},\quad\forall
i=1,\cdots,m,\ \ \text{as}\ \ \alpha\to\infty,$

that is, $\lambda_{i,\alpha}\to\lambda_{i}$ for all $i$, as we wanted, since
$\lambda_{i,\alpha}=\left(\frac{2}{\alpha+2}\right)^{2}\widehat{\Lambda}_{i,\alpha}$.
∎

######  Remark 6.2.

By the previous lemma,

$\frac{\widehat{\Lambda}_{i,\alpha}}{\alpha^{2}}\to\frac{\lambda_{i}}{4},\quad\alpha\to\infty,\quad\forall
i=1,\cdots,m,$

and therefore

$\widehat{\Lambda}_{i,\alpha}=\frac{\lambda_{i}}{4}\alpha^{2}+o(\alpha^{2}),\quad\alpha\to\infty,\quad
i=1,\cdots,m.$ (6.5)

###### Proof of Theorem 1.3.

We first recall that the Morse index of $u_{\alpha}$ is precisely the number
of negative eigenvalues (with their multiplicity) of (6.1). Moreover, each of
these negative eigenvalues is given (and conversely) by the decomposition
(6.2). Thus, for each $i=1,\cdots,m$, we need to know the numbers
$j\in\mathbb{N}$ that satisfy

$\widehat{\Lambda}_{i,\alpha}+j(N-2+j)<0.$ (6.6)

Using (6.5), the above inequality is equivalent to

$j<\frac{\sqrt{(N-2)^{2}-(N-2)\lambda_{i}\alpha^{2}+o(\alpha^{2})}-(N-2)}{2}.$

Since $0<-\lambda_{m}/2<-(N-2)\lambda_{i}$ for $N\geq 3$ and $i=1,\cdots,m$,
there exists $\alpha^{*}>\alpha_{p}$ such that

$J_{\alpha}\leq\frac{\sqrt{(N-2)^{2}-(\lambda_{m}/2)\alpha^{2}}}{2}+1<\frac{\sqrt{(N-2)^{2}-(N-2)\lambda_{i}\alpha^{2}+o(\alpha^{2})}-(N-2)}{2},$

for all $\alpha\geq\alpha^{*}$ and $i=1,\cdots,m$. Consequently, whenever
$j\leq J_{\alpha}$ with $\alpha\geq\alpha^{*}$, we get (6.6) for all
$i=1,\cdots,m$.

To show (1.2), observe that the radial eigenvalues of (6.1) are simple and the
numbers $j(N-2+j)$, with $j\in\mathbb{N}$, are eigenvalues of the Laplace-
Beltrami operator $-\Delta_{S^{N-1}}$ whose multiplicity is $N_{j}$. It
follows then, from [3, Proposition 4.1], that each eigenvalue
$\widehat{\Lambda}_{i,\alpha}+j(N-2+j)$ has multiplicity $N_{j}$. Therefore we
conclude that the linearized operator $L^{\alpha}$ has at least
$m\sum_{j=1}^{J_{\alpha}}N_{j}$ negative eigenvalues (with their multiplicity)
associated to nonradial eigenfunction for all $\alpha\geq\alpha^{*}$. Adding
the number of negative radial eigenvalues, which is $m$, we obtain (1.2).

Similarly, we can check (1.3). Indeed, given $\theta>1$, then
$0<-\lambda_{m-1}/\theta<-(N-2)\lambda_{i}$ for $N\geq 3$ and
$i=1,\cdots,m-1$. Thus, there exists
$\alpha^{\star}=\alpha^{\star}(\theta)>\alpha_{p}$ such that

$K_{\alpha}(\theta)<\frac{\sqrt{(N-2)^{2}-(N-2)\lambda_{i}\alpha^{2}+o(\alpha^{2})}-(N-2)}{2}\quad\forall\alpha\geq\alpha^{\star},\
\ i=1,\cdots,m-1.$

This implies that (6.6) holds for all $i=1,\cdots,m-1$, whenever $j\leq
K_{\alpha}(\theta)$ with $\alpha\geq\alpha^{\star}$, and hence (1.3) follows.
∎

To prove the monotonicity of the Morse indices with respect to $\alpha$ we
need to improve the asymptotic expansions (6.5). It is proved in [17] that the
functions
$\alpha\mapsto\widehat{\Lambda}_{i}(\alpha):=\widehat{\Lambda}_{i,\alpha}$,
$i=1,\cdots,m$, belong to $C^{1}(\alpha_{p},\infty)$ and satisfy

$\widehat{\Lambda}_{i}(\alpha)=\nu_{i}^{*}\alpha^{2}+c_{i}^{*}\alpha+o({\alpha})\quad\text{and}\quad\widehat{\Lambda}_{i}^{\prime}(\alpha)=2\nu_{i}^{*}\alpha+c_{i}^{*}+o(1),\quad\text{as}\
\alpha\to\infty,$ (6.7)

where $c_{i}^{*}$, $i=1,\cdots,m$, are constants and the values $\nu_{i}^{*}$
are the negative eigenvalues of a suitable one-dimensional eigenvalue problem;
see [17, Theorem 1.3]. In terms of the eigenvalues $\lambda_{i}$,
$i=1,\cdots,m$, we may write the asymptotic expansions (6.7) as follows. For
each $M\geq 2$ with $p(M-2)<M+2$, let $v_{M}$ be the unique solution with $m$
nodal sets of ($Q_{M}$) such that $v_{M}(0)>0$. Then the eigenvalue problem

$-\psi^{\prime\prime}-\frac{M-1}{t}\psi^{\prime}-p|v_{M}|^{p-1}\psi=\mu\frac{\psi}{t^{2}},\quad
t\in(0,1),\quad\psi\in H^{1}_{0,M},$ (6.8)

has exactly $m$ negative eigenvalues, say $\mu_{1}(M)<\cdots<\mu_{m}(M)<0$.
Observe that, putting $M=M_{\alpha}=\frac{2(\alpha+N)}{\alpha+2}$, the
problems (6.8) and (6.3) are the same. Thus

$\mu_{i}(M_{\alpha})=\lambda_{i,\alpha}=\left(\frac{2}{\alpha+2}\right)^{2}\widehat{\Lambda}_{i}(\alpha),$

and therefore, the maps $M\mapsto\mu_{i}(M)$ are $C^{1}$-functions for each
$i=1,\cdots,m$.

######  Proposition 6.3.

The $C^{1}$-functions $\alpha\mapsto\widehat{\Lambda}_{i}(\alpha)$,
$i=1,\cdots,m$, satisfy

$\widehat{\Lambda}_{i}(\alpha):=\widehat{\Lambda}_{i,\alpha}=\frac{\lambda_{i}}{4}\alpha^{2}+c_{i}\alpha+o({\alpha})\quad\text{and}\quad\widehat{\Lambda}_{i}^{\prime}(\alpha)=\frac{\lambda_{i}}{2}\alpha+c_{i}+o(1)\quad\text{as}\
\alpha\to\infty,$

where $c_{i}=\frac{(N-2)\mu_{i}^{\prime}(2)}{2}+\lambda_{i}$, $i=1,\cdots,m$.

###### Proof.

We may write

$\mu_{i}(M)=\mu_{i}(2)+\mu_{i}^{\prime}(2)(M-2)+o(M-2)\quad\text{and}\quad\mu_{i}^{\prime}(M)=\mu_{i}^{\prime}(2)+o(1)\quad\text{as
}M\to 2^{+}.$

Note that $M_{\alpha}\to 2^{+}$ as $\alpha\to\infty$. So, performing the
transformation $M\leftrightarrow M_{\alpha}$, a $o(M-2)$-function as $M\to
2^{+}$ corresponds to a $o(M_{\alpha}-2)=o(\frac{1}{\alpha})$-function as
$\alpha\to\infty$. Hence

$\mu_{i}(M_{\alpha})=\mu_{i}(2)+\frac{2(N-2)\mu_{i}^{\prime}(2)}{\alpha+2}+o\left(\frac{1}{\alpha}\right)\quad\text{and}\quad\mu_{i}^{\prime}(M_{\alpha})=\mu_{i}^{\prime}(2)+o(1)\quad\text{as
}\alpha\to\infty.$

Therefore

$\displaystyle\widehat{\Lambda}_{i}(\alpha)$
$\displaystyle=\left(\frac{\alpha+2}{2}\right)^{2}\mu_{i}(M_{\alpha})=\frac{\mu_{i}(M_{\alpha})}{4}\alpha^{2}+\mu_{i}(M_{\alpha})\alpha+\mu_{i}(M_{\alpha})$
$\displaystyle=\frac{\mu_{i}(2)}{4}\alpha^{2}+\frac{(N-2)\mu_{i}^{\prime}(2)}{2}\frac{\alpha^{2}}{\alpha+2}+\mu_{i}(2)\alpha+o(\alpha)$
$\displaystyle=\frac{\mu_{i}(2)}{4}\alpha^{2}+\left[\frac{(N-2)\mu_{i}^{\prime}(2)}{2}+\mu_{i}(2)\right]\alpha+o(\alpha),\quad\text{as
$\alpha\to\infty$},$

and

$\displaystyle\widehat{\Lambda}_{i}^{\prime}(\alpha)$
$\displaystyle=\frac{\alpha+2}{2}\mu_{i}(M_{\alpha})-\left(\frac{\alpha+2}{2}\right)^{2}\mu_{i}^{\prime}(M_{\alpha})\frac{2(N-2)}{(\alpha+2)^{2}}$
$\displaystyle=\frac{\alpha+2}{2}\mu_{i}(2)+(N-2)\mu_{i}^{\prime}(2)-\frac{(N-2)\mu_{i}^{\prime}(2)}{2}+o(1)$
$\displaystyle=\frac{\mu_{i}(2)}{2}\alpha+\mu_{i}(2)+\frac{(N-2)\mu_{i}^{\prime}(2)}{2}+o(1),\quad\text{as
}\alpha\to\infty.$

Finally, observing that $\mu_{i}(2)=\lambda_{i}$, we conclude the proof. ∎

###### Proof of Theorem 1.5.

It is a consequence of (6.2) and Proposition 6.3. Indeed, since
$\lambda_{i}<0$, by Proposition 6.3,

$\widehat{\Lambda}_{i}^{\prime}(\alpha)\to-\infty,\quad\alpha\to\infty,\quad\forall
i=1,\cdots,m.$

This shows in particular that there exists $\alpha_{*}>\alpha_{p}$ such that
$\widehat{\Lambda}_{i}^{\prime}(\alpha)<0$ for all $\alpha\geq\alpha_{*}$,
that is, the function $\alpha\mapsto\widehat{\Lambda}_{i,\alpha}$ is strictly
decreasing in $[\alpha_{*},\infty)$ for each $i=1,\cdots,m$. This implies,
using (6.2), that the Morse index $m(u_{\alpha})$ is nondecreasing with
respect to $\alpha$ in $[\alpha_{*},\infty)$. ∎

## References

  * [1] A. Aftalion and F. Pacella.  Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains.  C. R. Math. Acad. Sci. Paris, 339(5):339–344, 2004.
  * [2] A. L. Amadori and F. Gladiali.  Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem.  Calc. Var. Partial Differential Equations, 58(5):Paper No. 168, 47, 2019.
  * [3] A. L. Amadori and F. Gladiali.  On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE’s.  Nonlinear Anal. Real World Appl., 55:103133, 38, 2020.
  * [4] A. L. Amadori and F. Gladiali.  On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE’s: II.  Nonlinearity, 33(6):2541–2561, 2020.
  * [5] T. Bartsch, T. Weth, and M. Willem.  Partial symmetry of least energy nodal solutions to some variational problems.  J. Anal. Math., 96:1–18, 2005.
  * [6] J. Byeon and Z.-Q. Wang.  On the Hénon equation: asymptotic profile of ground states. II.  J. Differential Equations, 216(1):78–108, 2005.
  * [7] J. Byeon and Z.-Q. Wang.  On the Hénon equation: asymptotic profile of ground states. I.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 23(6):803–828, 2006.
  * [8] D. Cao, S. Peng, and S. Yan.  Asymptotic behaviour of ground state solutions for the Hénon equation.  IMA J. Appl. Math., 74(3):468–480, 2009.
  * [9] C. Cowan and N. Ghoussoub.  Estimates on pull-in distances in microelectromechanical systems models and other nonlinear eigenvalue problems.  SIAM J. Math. Anal., 42(5):1949–1966, 2010.
  * [10] F. De Marchis, I. Ianni, and F. Pacella.  A Morse index formula for radial solutions of Lane-Emden problems.  Adv. Math., 322:682–737, 2017.
  * [11] E. M. dos Santos and F. Pacella.  Hénon-type equations and concentration on spheres.  Indiana Univ. Math. J., 65(1):273–306, 2016.
  * [12] B. Gidas, W. M. Ni, and L. Nirenberg.  Symmetry and related properties via the maximum principle.  Comm. Math. Phys., 68(3):209–243, 1979.
  * [13] F. Gladiali, M. Grossi, and S. L. N. Neves.  Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane.  Commun. Contemp. Math., 18(5):1550087, 31, 2016.
  * [14] A. Harrabi, S. Rebhi, and A. Selmi.  Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index.  J. Differential Equations, 251(9):2409–2430, 2011.
  * [15] M. Hénon.  Numerical experiments on the stability of spherical stellar systems.  Astronomy and astrophysics, 24:229–238, 1973.
  * [16] R. Kajikiya.  Sobolev norms of radially symmetric oscillatory solutions for superlinear elliptic equations.  Hiroshima Math. J., 20(2):259–276, 1990.
  * [17] J. Kübler and T. Weth.  Spectral asymptotics of radial solutions and nonradial bifurcation for the henon equation.  Discrete Contin. Dyn. Syst., 40(6):3629–3656, 2020.
  * [18] W. Leite da Silva and E. Moreira dos Santos.  Monotonicity of the Morse index of radial solutions of the Hénon equation in dimension two.  Nonlinear Anal. Real World Appl., 48:485–492, 2019.
  * [19] Z. Lou, T. Weth, and Z. Zhang.  Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type.  Z. Angew. Math. Phys., 70(1):Paper No. 35, 19, 2019.
  * [20] E. Moreira dos Santos and F. Pacella.  Morse index of radial nodal solutions of Hénon type equations in dimension two.  Commun. Contemp. Math., 19(3):1650042, 16, 2017.
  * [21] K. Nagasaki.  Radial solutions for $\Delta u+|x|^{l}|u|^{p-1}u=0$ on the unit ball in ${\bf R}^{n}$.  J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36(2):211–232, 1989.
  * [22] W.-M. Ni.  Uniqueness of solutions of nonlinear Dirichlet problems.  J. Differential Equations, 50(2):289–304, 1983.
  * [23] D. Smets and M. Willem.  Partial symmetry and asymptotic behavior for some elliptic variational problems.  Calc. Var. Partial Differential Equations, 18(1):57–75, 2003.
  * [24] D. Smets, M. Willem, and J. Su.  Non-radial ground states for the Hénon equation.  Commun. Contemp. Math., 4(3):467–480, 2002.
  * [25] M. Willem.  Minimax Theorems.  Birkhäuser Boston, 1996.


The LHCspin project

M. Santimaria1$\star$, V. Carassiti2, G. Ciullo2,3, P. Di Nezza1, P.
Lenisa2,3, S. Mariani4,5, L. L. Pappalardo2,3 and E. Steffens6

1 INFN Laboratori Nazionali di Frascati, Frascati, Italy

2 INFN Ferrara, Italy

3 Department of Physics, University of Ferrara, Italy

4 INFN Firenze, Italy

5 Department of Physics, University of Firenze, Italy

6 Physics Dept., FAU Erlangen-Nurnberg, Erlangen, Germany

*<EMAIL_ADDRESS>

Presented at the Low-$x$ Workshop, Elba Island, Italy, September 27–October 1
2021

## Abstract

The goal of LHCspin is to develop, in the next few years, innovative solutions
and cutting-edge technologies to access spin physics in polarised fixed-target
collisions at high energy, exploring the unique kinematic regime offered by
LHC and exploiting new final states by means of the LHCb detector. The forward
geometry of the LHCb spectrometer is perfectly suited for the reconstruction
of particles produced in fixed-target collisions. This configuration, with
centre of mass energies ranging from
$\sqrt{s_{\rm{NN}}}=115\leavevmode\nobreak\ \rm{GeV}$ in $p-p$ interactions to
$\sqrt{s_{\rm{NN}}}=72\leavevmode\nobreak\ \rm{GeV}$ in heavy ion collisions,
allows to cover a wide backward rapidity region, including the poorly explored
high$-x$ regime. With the instrumentation of the proposed target system, LHCb
will become the first experiment simultaneously collecting unpolarised beam-
beam collisions at $\sqrt{s_{\rm{NN}}}=14\leavevmode\nobreak\ \rm{TeV}$ and
both unpolarised and polarised beam-target collisions. The status of the
project is presented along with a selection of physics opportunities.

##  1 Introduction

The LHC delivers proton and lead beams with an energy of
$7\leavevmode\nobreak\ \rm{TeV}$ and $2.76\leavevmode\nobreak\ \rm{TeV}$ per
nucleon, respectively, with world’s highest intensity. A short run with xenon
ions was also performed in 2017, while an oxygen beam is currently foreseen
for the Run 3 Citron:2018lsq . Fixed-target proton-gas collisions occur at a
centre of mass energy per nucleon of up to $115\leavevmode\nobreak\ \rm{GeV}$.
This corresponds to a centre of mass rapidity shift of
$y-y_{\rm{CM}}\approx\rm{arcsinh}(\sqrt{E_{\rm{N}}/2M_{\rm{N}}})=4.8$, so that
the LHCb acceptance $(2<\eta<5)$ covers the backward and central rapidities in
the centre of mass frame. Such a coverage offers an unprecedented opportunity
to investigate partons carrying a large fraction of the target nucleon
momentum, i.e. large Bjorken$-x$ values, corresponding to large and negative
Feynman$-x$ values111$x_{\rm{F}}\approx x_{1}-x_{2}$ where $x_{1}$ and $x_{2}$
are the Bjorken$-x$ values of the beam and target nucleon, respectively..

The LHCb detector Alves:2008zz  is a general-purpose forward spectrometer
specialised in detecting hadrons containing $c$ and $b$ quarks, and the only
LHC detector able to collect data in both collider and fixed-target mode. It
is fully instrumented in the $2<\eta<5$ region with a vertex locator (VELO), a
tracking system, two Cherenkov detectors, electromagnetic and hadronic
calorimeters and a muon detector. Fig. 1 shows a scheme of the upgraded LHCb
detector which is currently being installed for the Run 3, starting in 2022.

Figure 1: The Run 3 LHCb detector.

The fixed-target physics program at LHCb is active since the installation of
the SMOG (System for Measuring the Overlap with Gas) device Aaij:2014ida ,
enabling the injection of noble gases in the beam pipe section crossing the
VELO detector at a pressure of $\mathcal{O}(10^{-7})\leavevmode\nobreak\
\rm{mbar}$. Precise measurements of charm LHCb:2018ygc  and antiproton
LHCb:2018jry  production have been published based on $p-\rm{Ar}$ and
$p-\rm{He}$ fixed-target collisions. Fig. 2 shows the high-quality and low-
background samples collected in just one week of dedicated SMOG runs during
Run 2.

Figure 2: $J/\psi\to\mu^{+}\mu^{-}$ (left) and $D^{0}\to K^{-}\pi^{+}$ (right)
SMOG samples from LHCb:2018ygc .

With the SMOG2 upgrade LHCbCollaboration:2673690 , an openable gas storage
cell, shown in Fig. 3, has been installed in 2020 in front of the VELO.

Figure 3: The SMOG2 storage cell in the open (left) and closed (right)
configuration.

The cell boosts the target areal density by a factor of $8$ to $35$ depending
on the injected gas species. In addition, SMOG2 data will be collected in the
upcoming Run 3 with a novel reconstruction software allowing simultaneous
data-taking of beam-gas and beam-beam collisions. Very high tracking
efficiency is expected in the beam-gas interaction region, despite its
upstream position with respect to the VELO. Furthermore, beam-gas and beam-
beam vertices are well detached along the $z$ coordinate, as shown in Fig. 4.
SMOG2 will offer a rich physics program for the Run 3 and, at the same time,
allows to investigate the dynamics of the beam-target system, setting the
basis for future developments.
The LHCspin project Aidala:2019pit  aims at extending the SMOG2 physics
program in Run 4 (expected to start in 2028) and, with the installation of a
polarised gas target, to bring spin physics at LHC by exploiting the well
suited LHCb detector. A selection of physics opportunities accessible at
LHCspin is presented in Sec. 2, while the experimental setup is discussed in
Sec. 3.

##  2 Physics case

The physics case of LHCspin covers three main areas: exploration of the wide
physics potential offered by unpolarised gas targets, investigation of the
nucleon spin and heavy ion collisions.

Figure 4: Left: VELO track reconstruction efficiency for beam-gas (red) and
beam-beam (blue) primary vertices (PV) LHCB-FIGURE-2019-007 . Right: kinematic
coverage of LHCspin (orange) and other existing facilities.

###  2.1 Measurements with unpolarised gases

Similarly to SMOG2, LHCspin will allow the injection of several species of
unpolarised gases: $\rm{H}_{2}$, $\rm{D}_{2}$, He, $\rm{N}_{2}$, $\rm{O}_{2}$,
Ne, Ar, Kr and Xe. The impact of the gas on the LHC beam lifetime is
negligible: the luminosity loss due to collisions on hydrogen in the cell has
a characteristic time of around $2000$ days, whereas typical runs last for
$10-20$ hours.
Injecting unpolarised gases yields excellent opportunities to investigate
parton distribution functions (PDFs) in both nucleons and nuclei in the
large-$x$ and intermediate $Q^{2}$ regime (Fig. 4, right), which are
especially affected by lack of experimental data and impact several fields of
physics from QCD to astrophysics. For example, the large acceptance and high
reconstruction efficiency of LHCb on heavy flavour states enables the study of
gluon PDFs, which represent fundamental inputs for theoretical predictions
Hadjidakis:2018ifr  and are currently affected by large uncertainties, as
shown in the example of Fig. 5 (left). In addition, the structure of heavy
nuclei is known to depart from that obtained by the simple sum of free protons
and neutrons: within the unique acceptance of LHCb, a large amount of data can
be collected to shed light on the intriguing anti-shadowing effect expected at
$x\sim 0.1$ Eskola:2016oht , as shown in Fig. 5 (right).

Figure 5: Left: Relative uncertainty on the CT$14$ gluon PDF in a proton
Dulat:2015mca . Right: relative uncertainties on a set of gluon nuclear PDFs
in a lead nucleus Aidala:2019pit .

With the large amount of data to be collected with LHCspin, several
measurements impacting astrophysics and cosmic ray physics become possible.
For example, heavy-flavour hadroproduction directly impacts the knowledge of
prompt muonic neutrino flux Garzelli:2016xmx , which is especially affected by
PDF uncertainties, while large samples of proton collisions on helium, oxygen
and nitrogen provide valuable inputs to improve the understanding of the
compositions of ultra-high energy cosmic rays. Moreover, the possibility of
injecting an oxygen beam opens new and exciting prospects for antiproton
measurements Brewer:2021kiv .

###  2.2 Spin physics

Beside the colliner PDFs mentioned in Sec. 2.1, polarised quark and gluon
distributions can be probed at LHCspin by means of proton collisions on
polarised hydrogen and deuterium. Fig. 6 shows the 5D Wigner distributions
Bhattacharya:2017bvs  which, upon integration on the transverse momentum, lead
to the observable generalised parton distributions (GPDs), while transverse
momentum dependent distributions (TMDs) are obtained when integrating over the
transverse coordinate. There are several leading-twist distributions that can
be probed with unpolarised and transversely polarised quarks and nucleons,
giving independent information on the spin structure of the nucleon.

Figure 6: Wigner distributions (top) and leading-twist GPDs and TMDs for
different combinations of quark$\times$nucleon polarisation states (bottom).
Distributions marked in red vanish for no orbital angular momentum
contribution to the nucleon spin, while the quantities highlighted in green
can be accessed at LHCspin pasquini .

To access the transverse motion of partons within a polarised nucleon,
transverse single spin asymmetries (TSSAs) can be measured. For example

$A_{N}=\frac{1}{\it{P}}\frac{\sigma^{\uparrow}-\sigma^{\downarrow}}{\sigma^{\uparrow}+\sigma^{\downarrow}}\sim\frac{f^{q}_{1}(x_{1},k^{2}_{T1})\otimes
f^{\perp\overline{q}}_{1T}(x_{2},k^{2}_{T2})}{f^{q}_{1}(x_{1},k^{2}_{T1})\otimes
f^{\overline{q}}_{1}(x_{2},k^{2}_{T2})}$ (1)

in the polarised Drell-Yan (DY) channel probes the product of $f_{1}$
(unpolarised TMD) and $f^{\perp}_{1T}$ (Sivers function) in quarks and
antiquarks in the low $(x_{1})$ and high $(x_{2})$ $x$ regimes. Projections
for the uncertainty of such measurements are shown in Fig. 7 (left) based on
an integrated luminosity of $10\leavevmode\nobreak\ \rm{fb}^{-1}$. Being
T-odd, it is theoretically established that the Sivers function changes sign
in polarised DY with respect to semi-inclusive deep inelastic scattering
Collins:2002kn . This fundamental QCD prediction can be verified by exploiting
the large sample of DY data expected at LHCspin. In addition, isospin effects
can be investigated by comparing $p-\rm{H}$ and $p-\rm{D}$ collisions.
Several TMDs can be probed by evaluating the azimuthal asymmetries of the
produced dilepton pair: projected precisions for three such asymmetries are
shown in Fig. 7 (left).

Figure 7: Left: Measurements of $A_{N}$ in DY events as a function of $x$
compared to two theoretical predictions Aidala:2019pit . Right: projected
precision for some azimuthal asymmetry amplitudes with DY data as a function
of the dilepton invariant mass Hadjidakis:2018ifr .

Heavy flavour states will be the strength point of LHCspin. Being mainly
produced via gluon fusion at LHC, quarkonia and open heavy flavour states will
allow to probe the unknown gluon Sivers function via inclusive production of
$J/\psi$, $D^{0}$ but also with several unique states like $\eta_{c}$,
$\chi_{c}$, $\chi_{b}$ or $J/\psi J/\psi$. Fig. 8 (left) shows two predictions
for $A_{N}$ on $J/\psi$ events: $5-10\%$ asymmetries are expected in the
$x_{F}<0$ region, where the LHCspin sensitivity is the highest. Heavy flavour
states can be exploited as well to probe the gluon-induced asymmetries
$h^{\perp g}_{1}$ (Boer-Mulders) and $f_{1}^{g}$ (always present at the
denominator of $A_{N}$), which are both experimentally unconstrained.

Figure 8: Left: theoretical predictions for $A_{N}$ in inclusive $J/\psi$
production DAlesio:2018rnv . Right: up quark densities in momentum space
Bacchetta:2017gcc .

While TMDs provide a “tomography” of the nucleon in momentum space (Fig. 8,
right), a 3D picture in the spatial coordinates can be built by measuring
GPDs, as shown in Fig. 9.

Figure 9: Distortion of the up and down quark distributions in the impact
parameter space when spin is taken into account saggiatore .

Correlating position and momentum, GPDs also quantify the parton orbital
angular momentum, whose contribution to the total nucleon spin can be
inferred, for example, via the Ji sum rule Ji:1996ek . GPDs can be
experimentally probed with exclusive dilepton and quarkonia productions in
ultra-peripheral collisions, which are dominated by the electromagnetic
interaction. For example, TSSAs can be exploited to access the $E_{g}$
function, which has never been measured and represents a key element of the
proton spin puzzle. It is also attractive to measure the elusive transversity
PDF, whose knowledge is currently limited to valence quarks at the leading
order Radici:2018iag , as well as its integral, the tensor charge, which is of
direct interest in constraining physics beyond the Standard Model
Courtoy:2015haa .

###  2.3 Heavy ion collisions

Thermal heavy-flavour production is negligible at the typical temperature of
few hundreds MeV of the system created in heavy-ion collisions. Quarkonia
states ($c\overline{c}$, $b\overline{b}$) are instead produced on shorter
timescales, and their energy change while traversing the medium represents a
powerful way to investigate Quark-Gluon Plasma (QGP) properties. LHCb
capabilities allow to both cover the aforementioned charmonia and bottomonia
studies and to extend them to beauty baryons as well as to exotic probes. The
QGP phase diagram exploration at LHCspin can be performed with a rapidity scan
at a centre of mass energy of $\sqrt{s_{\rm{NN}}}=72\leavevmode\nobreak\
\rm{GeV}$ which is in-between those accessed at RHIC and SPS. In addition,
flow measurements will greatly benefit from the excellent identification
performance of LHCb on charged and neutral light hadrons.
The dynamics of small systems is an interesting topic joining heavy-ion
collisions and spin physics. In the spin $1$ deuteron nucleus, the nucleon
matter distribution is prolate for $j_{3}=\pm 1$ and oblate for $j_{3}=0$,
where $j_{3}$ is the projection of the spin along the polarisation axis. In
ultra-relativistic lead ion collisions on transversely polarised deuteron, the
deformation of the target deuteron can influence the orientation of the
fireball in the transverse plane, quantified by the ellipticity, as shown in
Fig. 10. The measurement proposed in Broniowski:2019kjo  can easily be
performed at LHCspin on minimum bias events thanks to the high-intensity LHC
beam.

Figure 10: Left: sketch of a ultra-relativistic collision of a lead nucleus
against a transversely polarised deuteron in two different angular momentum
projections. Right: ellipticity with respect to the polarisation axis as a
function of the collision centrality with LHCspin kinematics
Broniowski:2019kjo .

##  3 Experimental setup

The LHCspin experimental setup is in R&D phase and calls for the development
of a new generation polarised target. The starting point for this ambitious
task is the setup of the polarised target system employed at the HERMES
experiment Airapetian:2004yf  and comprises three main components: an Atomic
Beam Source (ABS), a Polarised Gas Target (PGT) and a diagnostic system. The
ABS consists of a dissociator with a cooled nozzle, a Stern-Gerlach apparatus
to focus the wanted hyperfine states, and adiabatic RF-transitions for setting
and switching the target polarisation between states of opposite sign. The ABS
injects a beam of polarised hydrogen or deuterium into the PGT, which is
located in the LHC primary vacuum. The PGT hosts a T-shaped openable storage
cell, sharing the SMOG2 geometry, and a compact superconductive dipole magnet,
as shown in Fig. 11. The magnet generates a $300\leavevmode\nobreak\ \rm{mT}$
static transverse field with a homogeneity of 10%, which is found to be
suitable to avoid beam-induced depolarisation Steffens:2019kgb . Studies for
the inner coating of the storage cell are currently ongoing, with the aim of
producing a surface that minimises the molecular recombination rate as well as
the secondary electron yield.

Figure 11: A drawing of the PGT with the magnet coils (orange) and the iron
return yoke (blue) enclosing the storage cell. The VELO vessel and detector
box are shown in green and grey, respectively.

In Fig. 12 (left), the PGT is shown in the same location of the SMOG2 cell, a
configuration that offers a large kinematic acceptance and does not require
additional detectors in LHCb. Fig. 12 (right) shows the efficiency to
reconstruct a primary vertex and both tracks in simulated
$\Upsilon\to\mu^{+}\mu^{-}$ events as a function of $x_{F}$ under three
locations of a $20\leavevmode\nobreak\ \rm{cm}$-long storage cell. The
simulation is performed within the GAUSS framework Clemencic:2011zza  with
upgrade LHCb conditions. New algorithms are currently being developed for the
Run 3 fixed-target reconstruction and are expected to sensibly improve the
current performance as well as to enable to record LHCspin data in parallel
with $p-p$ collisions LHCB-FIGURE-2019-007 .

Figure 12: Left: The PGT and the VELO vessel (green). Right: simulated
reconstruction efficiency for $\Upsilon\to\mu^{+}\mu^{-}$ events with three
different cell positions, the blue line corresponding to the SMOG2 location.

The diagnostic system continuously analyses gas samples drawn from the PGT and
comprises a target gas analyser to detect the molecular fraction, and thus the
degree of dissociation, and a Breit-Rabi polarimeter to measure the relative
population of the injected hyperfine states.
An instantaneous luminosity of $\mathcal{O}(10^{32})\leavevmode\nobreak\
\rm{cm}^{-2}\rm{s}^{-1}$ is foreseen for fixed-target $p-\rm{H}$ collisions in
Run 4, with a further factor $3-5$ increase for the high-luminosity LHC phase
from Run 5 (2032).

##  4 Conclusions

The fixed-target physics program at LHC has been greatly enhanced with the
recent installation of the SMOG2 gas storage cell at LHCb. LHCspin is the
natural evolution of SMOG2 and aims at installing a polarised gas target to
bring spin physics at LHC for the first time, opening a whole new range of
exploration. With strong interest and support from the international
theoretical community, LHCspin is a unique opportunity to advance our
knowledge on several unexplored QCD areas, complementing both existing
facilities and the future Electron-Ion Collider Accardi:2012qut .

#### Funding information

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement STRONG – 2020 - No
824093.

## References

  * (1) Z. Citron et al., Report from Working Group 5: Future physics opportunities for high-density QCD at the LHC with heavy-ion and proton beams, CERN Yellow Rep. Monogr. 7 (2019) 1159–1410, [arXiv:1812.06772].
  * (2) LHCb Collaboration, A. A. Alves, Jr. et al., The LHCb Detector at the LHC, JINST 3 (2008) S08005.
  * (3) LHCb Collaboration, R. Aaij et al., Precision luminosity measurements at LHCb, JINST 9 (2014), no. 12 P12005, [arXiv:1410.0149].
  * (4) LHCb Collaboration, R. Aaij et al., Measurement of Antiproton Production in ${\rm pHe}$ Collisions at $\sqrt{s_{NN}}=110$ GeV, Phys. Rev. Lett. 121 (2018), no. 22 222001, [arXiv:1808.06127].
  * (5) LHCb Collaboration, R. Aaij et al., First Measurement of Charm Production in its Fixed-Target Configuration at the LHC, Phys. Rev. Lett. 122 (2019), no. 13 132002, [arXiv:1810.07907].
  * (6) LHCb Collaboration, LHCb SMOG Upgrade, tech. rep., CERN, Geneva, May, 2019.
  * (7) C. A. Aidala et al., The LHCSpin Project, arXiv:1901.08002.
  * (8) LHCb Collaboration, SMOG2 Velo tracking efficiency, . https://cds.cern.ch/record/2688875.
  * (9) C. Hadjidakis et al., A Fixed-Target Programme at the LHC: Physics Case and Projected Performances for Heavy-Ion, Hadron, Spin and Astroparticle Studies, arXiv:1807.00603.
  * (10) K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Salgado, EPPS16: Nuclear parton distributions with LHC data, Eur. Phys. J. C 77 (2017), no. 3 163, [arXiv:1612.05741].
  * (11) S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, and C. P. Yuan, New parton distribution functions from a global analysis of quantum chromodynamics, Phys. Rev. D 93 (2016), no. 3 033006, [arXiv:1506.07443].
  * (12) PROSA Collaboration, M. V. Garzelli, S. Moch, O. Zenaiev, A. Cooper-Sarkar, A. Geiser, K. Lipka, R. Placakyte, and G. Sigl, Prompt neutrino fluxes in the atmosphere with PROSA parton distribution functions, JHEP 05 (2017) 004, [arXiv:1611.03815].
  * (13) J. Brewer, A. Mazeliauskas, and W. van der Schee, Opportunities of OO and $p$O collisions at the LHC, in Opportunities of OO and pO collisions at the LHC, 3, 2021.  arXiv:2103.01939.
  * (14) S. Bhattacharya, A. Metz, and J. Zhou, Generalized TMDs and the exclusive double Drell–Yan process, Phys. Lett. B 771 (2017) 396–400, [arXiv:1702.04387]. [Erratum: Phys.Lett.B 810, 135866 (2020)].
  * (15) B. Pasquini, “Recent advances on spin and 3d nucleon structure.” https://indico.bnl.gov/event/9726/contributions/47605/attachments/33594/54332/DIS2021-Pasquini.pdf.
  * (16) J. C. Collins, Leading twist single transverse-spin asymmetries: Drell-Yan and deep inelastic scattering, Phys. Lett. B 536 (2002) 43–48, [hep-ph/0204004].
  * (17) U. D’Alesio, C. Flore, F. Murgia, C. Pisano, and P. Taels, Unraveling the Gluon Sivers Function in Hadronic Collisions at RHIC, Phys. Rev. D 99 (2019), no. 3 036013, [arXiv:1811.02970].
  * (18) A. Bacchetta, F. Delcarro, C. Pisano, M. Radici, and A. Signori, Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production, JHEP 06 (2017) 081, [arXiv:1703.10157]. [Erratum: JHEP 06, 051 (2019)].
  * (19) A. Bacchetta and M. Contalbrigo, The proton in 3D, Il Nuovo Saggiatore 28 (2012) 16–27.
  * (20) X.-D. Ji, Gauge-Invariant Decomposition of Nucleon Spin, Phys. Rev. Lett. 78 (1997) 610–613, [hep-ph/9603249].
  * (21) M. Radici and A. Bacchetta, First Extraction of Transversity from a Global Analysis of Electron-Proton and Proton-Proton Data, Phys. Rev. Lett. 120 (2018), no. 19 192001, [arXiv:1802.05212].
  * (22) A. Courtoy, S. Baeßler, M. González-Alonso, and S. Liuti, Beyond-Standard-Model Tensor Interaction and Hadron Phenomenology, Phys. Rev. Lett. 115 (2015) 162001, [arXiv:1503.06814].
  * (23) W. Broniowski and P. Bożek, Elliptic flow in ultrarelativistic collisions with light polarized nuclei, Phys. Rev. C 101 (2020), no. 2 024901, [arXiv:1906.09045].
  * (24) HERMES Collaboration, A. Airapetian et al., The HERMES polarized hydrogen and deuterium gas target in the HERA electron storage ring, Nucl. Instrum. Meth. A 540 (2005) 68–101, [physics/0408137].
  * (25) E. Steffens, V. Carassiti, G. Ciullo, P. Di Nezza, P. Lenisa, L. L. Pappalardo, and A. A. Vasilyev, Design Consideration on a Polarized Gas Target for the LHC, PoS SPIN2018 (2019) 098, [arXiv:1901.06361].
  * (26) LHCb Collaboration, M. Clemencic, G. Corti, S. Easo, C. R. Jones, S. Miglioranzi, M. Pappagallo, and P. Robbe, The LHCb simulation application, Gauss: Design, evolution and experience, J. Phys. Conf. Ser. 331 (2011) 032023.
  * (27) A. Accardi et al., Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all, Eur. Phys. J. A 52 (2016), no. 9 268, [arXiv:1212.1701].