# On the mixed local–nonlocal Hénon equation

Ariel M. Salort  and  Eugenio Vecchi  Instituto del Cálculo Luis A. Santaló
(IMAS), CONICET, Departamento de Matemática, FCEN - Universidad de Buenos
Aires Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n., Buenos Aires,
Argentina<EMAIL_ADDRESS>http://mate.dm.uba.ar/ asalort Dipartimento di
Matematica
Politecnico di Milano
Via Bonardi 9, 20133 Milano, Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider a Hénon-type equation driven by a nonlinear operator
obtained as a combination of a local and nonlocal term. We prove existence and
non-existence akin to the classical result by Ni, and a stability result as
the fractional parameter $s\to 1$.

###### 2010 Mathematics Subject Classification:

35R11, 35B35

##  1\. Introduction

Given $\beta\in[0,1]$, a fractional parameter $s\in(0,1)$ and $p>1$ we
consider the following mixed local–nonlocal elliptic operator

(1.1)
$\mathcal{L}_{s,p,\beta}u:=(1-\beta)(-\Delta_{p})u+\beta(-\Delta_{p})^{s}u.$

It is obvious that $\mathcal{L}_{s,p,\beta}$ boils down to the usual
$p-$Laplacian operator when $\beta=0$ and to the pure fractional $p-$Laplacian
when $\beta=1$. In the linear case $p=2$, this operator has a probabilistic
interpretation consisting in describing a discontinuous process where the
local part provides the continuous part, while the nonlocal one represents the
jump process. We postpone to the end of the Introduction a brief account of
the existing literature.

In this paper we consider the following Dirichlet boundary value problem

(1.2)
$\left\\{\begin{array}[]{rll}\mathcal{L}_{s,p,\beta}u=|x|^{\alpha}u^{q-1}&\text{in
$B$}&\\\ u>0&\text{in $B$}&\\\ u=0&\text{on $\partial B$}&\text{if
$\beta=0$}\\\ u=0&\text{in $\mathbb{R}^{n}\setminus B$}&\text{if
$\beta\in(0,1]$,}\end{array}\right.$

where $B$ stands for the unit ball in $\mathbb{R}^{n}$, $n\geq 2$, $\alpha>0$,
$s\in(0,1)$, $p>1$; we assume that $n>sp$ if $\beta=1$, while $n>p$ if
$\beta\in[0,1)$, and $p<q<p^{*}_{\beta}$, where

$p^{*}_{\beta}:=\left\\{\begin{array}[]{rl}\frac{np}{n-p}&\textrm{ if
}\beta\in[0,1)\\\ \frac{np}{n-sp}&\textrm{ if }\beta=1.\end{array}\right.$

Here we are primarily interested in proving existence and non-existence of
weak solutions, which we define in a classical variational way: we consider
the functional $\mathcal{J}\colon X_{\beta,rad}(B)\to\mathbb{R}$ associated to
(1.2)

(1.3) $\mathcal{J}(u)=\frac{1-\beta}{p}\|\nabla
u\|^{p}_{L^{p}(B)}+\frac{\beta}{p}[u]_{s,p}^{p}-\frac{1}{q}\int_{B}|x|^{\alpha}(u^{+})^{q}\,dx,$

defined on a suitable space $X_{\beta,rad}(B)$, see Section 2.1, where
$[u]_{s,p}$ stands for the Gagliardo seminorm

(1.4)
$[u]_{s,p}:=K(n,s)\,\left(\iint_{\mathbb{R}^{2n}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{n+2s}}\,dx\,dy\right)^{1/p},$

where $K(n,s)$ is a positive constant, depending only on $n$ and $s$, such
that $[u]_{s,p}^{p}\to\|\nabla u\|_{p}^{p}$ as $s\to 1$, see e.g. [7].
Clearly, $\mathcal{J}$ is class $C^{1}$ and its Frechét derivative is given by
the expression

$\displaystyle\langle\mathcal{J}^{\prime}(u),v\rangle$
$\displaystyle=(1-\beta)\int_{B}\|\nabla u\|^{p-2}\langle\nabla u,\nabla
v\rangle\,dx$ $\displaystyle+\beta
K(n,s)\iint_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+sp}}\,dx\,dy$
$\displaystyle-\int_{B}|x|^{\alpha}(u^{+})^{q-1}v\,dx$

for all $v\in X_{\beta,rad}(B)$. Therefore, any critical point of
$\mathcal{J}$ is a weak solution of (1.2).

Obviously, when $\beta=0$ and $p=2$, problem (1.2) boils down to the classical
Hénon equation introduced in [29] to model spherically symmetric stellar
clusters. The literature related to this famous equation is huge and
encompasses several interesting lines of research in Nonlinear Analysis,
including existence of solutions, nonexistence, multiplicity and finer
qualitative properties of solutions. Here we do not aim at providing a
complete and fully detailed list of references, but rather we limit ourselves
in mentioning the papers which are closely related to the content of this
note. Our main interest is to extend a classical result in [32], where Ni
noticed that the presence of the term $|x|^{\alpha}$ was modifying the problem
enough to increase the range of powers of $u$ for which a solution exists, in
this way presenting a quite different scenario with respect to the case
$\alpha=0$. Indeed, he was able to show that in there is a weak solution for
$2<q<\tfrac{2n+2\alpha}{n-2}=2^{\ast}+\tfrac{2\alpha}{n-2}$, so going beyond
the classical nonexistence threshold $2^{\ast}$ related to the critical
Sobolev embedding. Interestingly, due to the method for proving such
existence, Ni was also able to get radiality of such a solution. Indeed, that
solution is of Mountain Pass type, exploiting the Radial Lemma of Strauss and
the compactness of radial Sobolev functions. This fact is noteworthy because
the term $|x|^{\alpha}$ prevents from applying the symmetry results due to
Gidas-Ni-Nirenberg already in the pure local case, and so even more in the
mixed case, where similar qualitative results have been recently proved in
[4]. Actually, already in the pure local case, in [38] it was proved that
there is a sort of critical threshold for the parameter $\alpha$ beyond which
there exist non-radial ground states.

Following the result of Ni, there have been several extension of it to
different operators. Let us briefly recall them. The case of the $p-$Laplacian
was treated in [31] for $p>1$ while the purely linear nonlocal case (i.e.,
$\beta=1$ and $p=2$) was addresses in [37] , where it was proved that for
$1<2s<n$ and $q<\frac{2n+2\alpha}{n-2s}$, there exists a positive weak
solution for (1.2). The critical case $q=2^{*}_{s,\alpha}$ was studied in [2].
To the best of our knowledge, the pure nonlocal and nonlinear case (i.e.
$\beta=1$ and $p\neq 2$) has not be covered so far: this is the main reason
why we choose the operator in (1.1), so to get existence and non-existence of
solutions for both the nonlocal nonlinear case as well as mixed local-nonlocal
combinations.

Before stating our results, let us introduce a further threshold quantity
which will play a major role in the following:

(1.5) $p^{*}_{\beta,\alpha}:=\left\\{\begin{array}[]{rl}\frac{np+\alpha
p}{n-p}&\textrm{ if }\beta\in[0,1)\\\ \frac{np+\alpha p}{n-sp}&\textrm{ if
}\beta=1.\end{array}\right.$

Our first results concern the range of existence and non-existence of weak
solutions to (1.2).

######  Theorem 1.1.

Let $\alpha>0$, $p>1$ and $s\in(0,1)$ be such that

$p<n\quad\text{ when }\beta\in[0,1),\qquad sp<n\quad\text{ when }\beta=1.$

Then for all $p<q<p^{*}_{\beta,\alpha}$ there exists a positive weak solution
$u\in X_{\beta,rad}(B)$ to (1.2).

Moreover, we have that $u\in L^{\infty}(B)$ for $\beta\in[0,1)$ when
$0<s<1<p<q<p^{*}_{\beta,\alpha}$ such are that $p<n$ and

(1.6) $\alpha>\max\left\\{0,(q-1)\left(\dfrac{n}{p}-1\right)-p\right\\},$

and for $\beta=1$ when $0<s<1<p<q<p^{*}_{\beta,\alpha}$ are such that $sp<n$
and

(1.7) $\alpha>\max\left\\{0,(q-1)\left(\dfrac{n}{p}-s\right)-sp\right\\},$

or for all $\alpha>0$ when $1<p<q<p^{*}_{\beta,\alpha}$ are such that $sp<n$
and the range for $s$ is

(1.8) $0<s<\frac{n}{p}\frac{q-1}{p+q-1}.$

######  Theorem 1.2.

Let $\alpha>0$, $p>1$ and $s\in(0,1)$ be such that

$p<n\quad\text{ when }\beta\in[0,1),\qquad sp<n\quad\text{ when }\beta=1.$

Then for all $q>p^{*}_{\beta,\alpha}$ problem (1.2) has no solutions $u\in
X_{\beta}(B)\cap W^{1,r}(B)\cap L^{\infty}(B)$, for some $r>1$.

In particular, there are no positive weak solutions to (1.2) for

  * •

$\beta\in[0,1)$ when $p<n$ and (1.6) holds;

  * •

$\beta=1$ when $sp<n$ and (1.7) holds, or when $\alpha>0$ and (1.8) holds.

The proof of Theorem 1.1 follows the scheme introduced by Ni in [32]. To this
aim, we need several ingredients, like a proper version of the Radial Lemma
(see Section 2.3) and compactness of the embeddings of the appropriate radial
Sobolev spaces. The proof of Theorem 1.2 is heavily based on a previous non-
existence result by Ros–Oton and Serra [34]. In order to apply it, we have to
prove a regularity result of independent interest, namely Theorem 2.4, where
we show the boundedness of weak solutions of

$\begin{cases}\mathcal{L}_{s,p,\beta}u=f&\text{in $\Omega$},\\\ u=0&\text{in
}\mathbb{R}^{n}\setminus\Omega,\end{cases}$

when $f\in L^{r}(\Omega)$ with $r>\tfrac{n}{p}$.

Our next result establishes the stability of solutions of uniformly bounded
solutions of (1.2) as $s\nearrow 1$.

######  Theorem 1.3.

Let $\beta\in(0,1]$. Let $p>1$ and $s_{k}\in(0,1)$ be such that $s_{k}p<n$ and
$s_{k}\to 1$ as $k\to+\infty$. Given $p<q<p^{*}_{\beta,\alpha}$, let $u_{k}\in
X_{\beta,rad}(B)$ be a solution of (1.2) such that

$\sup_{k\in\mathbb{N}}\,\|u_{k}\|_{\beta}<+\infty.$

Then, every accumulation point $u$ of $\\{u_{k}\\}_{k\in\mathbb{N}}$ in the
$L^{p}(B)$-topology is a weak solution of (1.2) with $\beta=0$.

Since the $\beta$-norm of weak solutions to (1.2) can be bounded uniformly on
$s$ (see Remark 3.3), we immediately obtain the following consequence.

######  Corollary 1.4.

Let $\beta\in(0,1]$. Let $p>1$ and $s_{k}\in(0,1)$ be such that $s_{k}p<n$ and
$s_{k}\to 1$ as $k\to+\infty$. Given $p<q<p^{*}_{\beta,\alpha}$, let $u_{k}\in
X_{\beta,rad}(B)$ be a weak solution of (1.2). Then

$u_{k}\to u\text{ strongly in }L^{p}(B)\text{ as }k\to\infty$

where $u\in X_{\beta,rad}(B)$ is a solution of (1.2) with $\beta=0$.

We remark that, since the above convergence implies in particular that
$u_{k}\to u$ a.e., the limit function $u$ is actually radial.

Our last result extend the result of Smets, Willem and Su to the linear mixed
case, showing the possibly intuitive fact that coupling a nonlocal part with
the local one still allows for non-radial solutions to (1.2).

######  Theorem 1.5.

Given $n\geq 3$ and $0\leq\beta<1$, then for any $2+\beta<q<2^{*}$, there
exists $\alpha^{*}>0$ such that no ground state of (1.2) with $p=2$ is radial,
provided $\alpha>\alpha^{*}$.

We want to stress that the proof exploits several relations between the local
and nonlocal terms and is heavily based on the original proof in [38] in the
linear case. This is the reason why the pure nonlocal case is actually not
covered by our result.

We want close this Introduction spending a few words regarding the existing
literature: as far as we know, Fonduun [23] was the first who proved bounds
for the heat kernel associated to (1.1) for $p=2$, a Harnack inequality and a
regularity theorem for mixed local-nonlocal harmonic functions. A few years
later, in [10] the authors proved Harnack estimates in the linear case.
Further regularity results for $p=2$ are contained in [3], where the authors
prove local $H^{k+2}$ estimates and several maximum principles. Qualitative
properties of solutions of semilinear equations in the spirit of the classical
results by Gidas, Ni and Nirenberg have been proved in [4], while a
quantitative version of a Faber-Krahn inequality has been proved in [5]. We
refer also to [28] for related results. Recently, the nonlinear operator (1.1)
has also been considered: in [9] the authors considered an eigenvalue problem
for a system of local-nonlocal operators and studied the asymptotics as
$p\to+\infty$. In a similar spirit, in [13] the authors considered a nonlinear
equation with concave-convex right hand side. We also quote [6] where the
authors considered a nonlinear version of the famous Brezis-Oswald problem,
and [24, 26] where the authors proved regularity results and studied mixed
Sobolev inequalities and quasilinear singular problems in the spirit of
Boccardo and Murat. We finally mention [17] for a first study of such
operators with (nonlocal) Neumann boundary conditions. We want also to remind
that there is a pretty active line of research dealing with evolution
equations having the operator (1.1) as elliptic part: we refer e.g. to [18,
25] and to [19] where such operators are used to propose a model to describe
the diffusion of a biological population living in an ecological niche and
subject to both local and nonlocal dispersals. Finally, we want to mention
that several papers, see e.g. [11, 15, 20, 27], have also investigated the
coupling of a local operator with a nonlocal and nonsingular one having a
different kernel with respect to the one considered in the present note.

The paper is organized as follows: in Section 2 we collect all the
preliminaries needed to prove our results. In particular, we introduce the
functional and variational setting to study (1.2), we collect the different
versions of the radial Lemma needed for our purposes and we prove the
boundedness of the solutions. In Section 3 we prove Theorem 1.1 and Theorem
1.2, while the proof of Theorem 1.3 is postponed to Section 4. In Section 5 we
prove Theorem 1.5. Finally, we add a brief Appendix A where we recall the
classical Mountain Pass Lemma and a few other classical results used along the
paper.

##  2\. Preliminaries

###  2.1. Function spaces

Problem (1.2) is the local-nonlocal and nonlinear counterpart of the Hénon
problem studied in the celebrated paper by Ni [32]. As already mentioned in
the Introduction, the problem has to be settled in the proper function space,
which we consider to be

(2.1) $X_{\beta}(B):=\left\\{\begin{array}[]{rl}W^{1,p}_{0}(B)&\textrm{if
}\beta=0\\\ X_{p}(B)&\textrm{if }\beta\in(0,1)\\\ W^{s,p}_{0}(B)&\textrm{if
}\beta=1,\end{array}\right.$

where $W^{1,p}_{0}(B)$ is the classical Sobolev space whose functions have
null trace on the boundary of $B$, while for $\beta=1$ we have the classical
fractional Sobolev space

(2.2) $W^{s,p}_{0}(B):=\left\\{u\in W^{s,p}(\mathbb{R}^{n}):u=0\textrm{ in
}\mathbb{R}^{n}\setminus B\right\\}.$

Concerning the true mixed local-nonlocal case, i.e. for $\beta\in(0,1)$, the
space $X_{p}(B)$ has been introduced in [6] for more general open sets
$\Omega\subset\mathbb{R}^{n}$ with $C^{1}$-smooth boundary. Nevertheless, to
simplify the reading, we recall its precision definition in our specific case:

(2.3) $X_{p}(B):=\left\\{u\in W^{1,p}(\mathbb{R}^{n}):\,\text{$u\equiv 0$
a.e.\,on $\mathbb{R}^{n}\setminus B$}\right\\}.$

Since $\partial B$ is smooth, we can identify $X_{p}(B)$ with the space
$W_{0}^{1,p}(B)$ in the following sense: denoting by $\mathbf{1}_{B}$ the
indicator function of the ball $B$, we have that

(2.4) $u\in W_{0}^{1,p}(B)\,\,\Longleftrightarrow\,\,u\cdot\mathbf{1}_{B}\in
X_{p}(B).$

Therefore, in what follows we will always identify a function $u\in
W_{0}^{1,p}(B)$ with $\hat{u}:=u\cdot\mathbf{1}_{B}\in X_{p}(B),$ obtained as
a zero-extension outside of $B$. We further notice that by the Poincaré
inequality and (2.4), we get that the quantity

$\|u\|_{X_{p}}:=\left(\int_{B}\|\nabla u\|^{p}\,dx\right)^{1/p},\qquad u\in
X_{p}(B),$

endows $X_{p}(B)$ with a structure of real Banach space, which is actually
isometric to $W_{0}^{1,p}(B)$. We briefly list a couple of useful properties
that hold true:

  1. (i)

$X_{p}(B)$ is separable and reflexive (being $p>1$);

  2. (ii)

$C_{0}^{\infty}(B)$ is dense in $X_{p}(B)$.

Summarizing, we will denote

(2.5) $\|u\|_{\beta}:=\left\\{\begin{array}[]{rl}\|\nabla
u\|_{L^{p}(B)}&\textrm{ if }\beta\in[0,1)\\\ \left[u\right]_{s,p}&\textrm{ if
}\beta=1.\end{array}\right.$

Finally, since we are mainly interested in proving results related to radial
functions, we introduce the following subspaces:

(2.6) $X_{\beta,rad}(B):=\\{u\in X_{\beta}(B):u\textrm{ is radial}\\}.$

###  2.2. The local-nonlocal operator

Given $\beta\in[0,1]$, a fractional parameter $s\in(0,1)$ and $p>1$ the mixed
local–nonlocal elliptic operator

$\mathcal{L}_{s,p,\beta}u:=(1-\beta)(-\Delta_{p})u+\beta(-\Delta_{p})^{s}u$

is well defined between $X_{\beta}(B)$ and its dual space $X^{*}_{\beta}(B)$
and the following representation formula holds:

$\displaystyle\langle\mathcal{L}_{s,p,\beta}u,v\rangle=(1-\beta)$
$\displaystyle\int_{B}|\nabla u|^{p-2}\langle\nabla u,\nabla v\rangle\,dx$
$\displaystyle+\beta
K(n,s)\iint_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+sp}}\,dx\,dy$

for any $v\in X_{\beta}(\mathbb{R}^{n})$.

Therefore, we say that $u\in X_{\beta}(B)$ is a _weak solution_ of (1.2) if

$\langle\mathcal{L}_{s,p,\beta}u,v\rangle=\int_{B}|x|^{\alpha}u^{q-1}v\,dx\qquad\textrm{for
all }v\in X_{\beta}(B).$

Observe that with this notation we have

(2.7)
$\langle\mathcal{L}_{s,p,\beta}u,v\rangle=\langle\mathcal{J}^{\prime}(u),v\rangle+\int_{\Omega}|x|^{\alpha}u^{q-1}v\,dx\quad\textrm{for
all }v\in X_{\beta}(B).$

######  Remark 2.1.

Given $u,v\in X_{\beta}(B)$, by Hölder’s inequality

$\displaystyle\langle\mathcal{L}_{s,p,\beta}u,v\rangle$
$\displaystyle\leq(1-\beta)\left(\int_{B}(|\nabla
u|^{p-1})^{p^{\prime}}\,dx\right)^{\frac{1}{p^{\prime}}}\left(\int_{B}|\nabla
v|^{p}\right)^{\frac{1}{p}}$ $\displaystyle+\beta
K(n,s)\left(\iint_{\mathbb{R}^{2n}}\left(\frac{|u(x)-u(y)|^{p-1}}{|x-y|^{\frac{n+sp}{p^{\prime}}}}\right)^{p^{\prime}}\,dxdy\right)^{\frac{1}{p^{\prime}}}\left(\iint_{\mathbb{R}^{2n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\right)^{\frac{1}{p}}$
$\displaystyle=(1-\beta)\|\nabla u\|_{L^{p}(\mathbb{R}^{n})}^{p-1}\|\nabla
v\|_{L^{p}(\mathbb{R}^{n})}+\beta[u]_{s,p}^{p-1}[v]_{s,p}.$

This relation together with [14, Proposition 2.2] gives that

$\langle\mathcal{L}_{s,p,\beta}u,v\rangle\leq
C\|u\|_{\beta}^{p-1}\|v\|_{\beta}\quad\textrm{for all }u,v\in
X_{\beta}(B)\text{ and for every }\beta\in[0,1].$

Therefore, in light of Proposition A.4, $\mathcal{L}_{s,p,\beta}$ satisfies
the so-called _$({\bf S})$ -property_ of compactness (see Definition A.3).

###  2.3. Radial Lemma

In [39], Strauss shed some light on the relation between the regularity and
the decay of a Sobolev function in $H^{1}(\mathbb{R}^{n})$. In particular, he
proved a nowadays famous pointwise inequality, often referred to as Strauss’
Radial Lemma, which reads as follows: let $n\geq 2$ and let $u\in
H^{1}_{rad}(\mathbb{R}^{n})$, then

(2.8) $|u(x)|\leq
C(n)|x|^{(1-n)/2}\left\\{\|u\|^{1/2}_{L^{2}(\mathbb{R}^{n})}\|\nabla
u\|^{1/2}_{L^{2}(\mathbb{R}^{n})}\right\\},\quad\textrm{for a.e.
}x\in\mathbb{R}^{n}.$

It is also possible to prove that there exists a radial function $\tilde{u}$
which coincides almost everywhere with $u$ and that it is continuous outside
of the origin. The inequality (2.8) plays a crucial role in proving compact
embeddings. According to [30, Remark I.3], the above inequality holds
replacing the exponent of $|x|$ with $-\alpha+\tfrac{2-n}{n}$ for every
$\alpha\in\left[0,\tfrac{1}{2}\right]$. Now, the case $\alpha=0$ gives back
the exponent found by Ni in [32], where he was dealing with radial functions
in $H^{1}_{0}(B)$. The nonlinear case is treated in [30], where Lions proved
the following: let $n\geq\max\\{2,p\\}$ and let $u\in
H^{1,p}_{rad}(\mathbb{R}^{n})$, then,

(2.9) $|u(x)|\leq
C(n,p)|x|^{-\gamma}\left\\{\|u\|^{(p-1)/p}_{L^{p}(\mathbb{R}^{n})}\|\nabla
u\|^{1/p}_{L^{p}(\mathbb{R}^{n})}\right\\},\quad\textrm{for a.e.
}x\in\mathbb{R}^{n},$

for every $\gamma=\tfrac{n}{p}-\delta$, where
$\delta\in\left[\tfrac{1}{p},1\right]$. The case $\delta=1$ gives the
analogous of [32] for a general $p\neq 2$.
For the nonlocal case, we refer to [12] in the linear case (see also [37]),
and [36] for the case $p\neq 2$.

In order to simplify the readability, we list below the variants of the
Strauss’ Lemma needed for our purposes. We start from the pure nonlocal case.

######  Lemma 2.2 ([36]).

Let $p>1$ with $1\leq sp<n$. Then there exists $C>0$ with

$|u(x)|\leq C|x|^{s-\frac{n}{p}}\|u\|_{W^{s,p}(\mathbb{R}^{n})}$

for every $u\in W^{s,p}_{rad}(\mathbb{R}^{n})$ and for a.e. $0<|x|\leq 1$.

In the mixed case, the radial lemma can be stated as follows:

######  Lemma 2.3.

Let $p\geq 1$ and let $n\geq\max\\{2,p\\}$. Then, for every radial function
$u\in X_{p,rad}(B)$, there exists a positive constant $C=C(n,p,\Omega)>0$ such
that

(2.10) $|u(x)|\leq C|x|^{1-\frac{n}{p}}\|\nabla
u\|_{L^{p}(B)},\quad\textrm{for a.e. }0<|x|\leq 1.$

###### Proof.

We first notice that by the very definition of the (radial) Sobolev space
$X_{p,rad}(B)$, and thanks to the regularity of $\partial B$, it actually
holds that $u\in W^{1,p}(\mathbb{R}^{n})$. Therefore, we can apply [30, Lemma
II.1, Remark II.3], finding that

(2.11) $|u(x)|\leq
C(n,p)|x|^{(p-n)/p}\left\\{\|u\|^{(p-1)/p}_{L^{p}(\mathbb{R}^{n})}\|\nabla
u\|^{1/p}_{L^{p}(\mathbb{R}^{n})}\right\\},\quad\textrm{for a.e.
}x\in\mathbb{R}^{n}.$

In particular, the latter holds for a.e. $0<|x|\leq 1$. Now, it suffices to
notice that

(2.12) $\displaystyle\|u\|_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle=\|u\|_{L^{p}(B)}\leq C(n,p,\Omega)\|\nabla
u\|_{L^{p}(B)}\quad\textrm{(Poincar\'{e} ineq.)}$ $\displaystyle\leq
C(n,p,\Omega)\|\nabla u\|_{L^{p}(\mathbb{R}^{n})}.$

Now, (2.10) easily follows. ∎

It remains the pure local case corresponding to $\beta=0$. As for the
considerations made on the space $X_{p}(B)$, thanks to the regularity of
$\partial B$, we can reverse the identification of $X_{p}(B)$ with
$W^{1,p}_{0}(B)$ and use Lemma 2.3 also in the case $\beta=0$.

We stress that the radial Strauss-Ni’s Lemmas presented before are the key
ingredients for the validity of the Palais-Smale condition.

###  2.4. Boundedness

We want now to discuss the boundedness of the weak solutions of (1.2). In the
case $\beta\in[0,1)$, we start with a general theorem of classical flavor
whose proof is an adaptation of the classical method by Stampacchia. This
approach has been already extended to the purely nonlocal setting, see e.g.
the proof of Proposition 9 in [35] and of Theorem 2.3 in [16], and even to the
linear mixed local-nonlocal case in [3]. Nevertheless, we will show all the
details in order to make the paper self-contained.
In the purely nonlocal case $\beta=1$, we refer to [8, Theorem 3.1].

######  Theorem 2.4.

Assume that $n\geq 3$ and that $\beta\in[0,1)$. Let $f\in L^{r}(\Omega)$, with
$1<p<n$, $r>n/p$, and assume that there exists the weak solution $u_{f}\in
X_{\beta}(\Omega)$ of

(2.13) $\begin{cases}\mathcal{L}_{s,p,\beta}u=f&\text{in $\Omega$},\\\
u=0&\text{in }\mathbb{R}^{n}\setminus\Omega.\end{cases}$

Then, $u_{f}\in L^{\infty}(\mathbb{R}^{n})$.

###### Proof of Theorem 2.4.

We note first that the case $\beta=0$ which corresponds to the purely local
case is already well-known.
Let $\delta>0$ be a positive number that we will conveniently choose later on.
We can certainly assume that $u_{f}\not\equiv 0$ else there is nothing to
prove. Therefore, we can define the functions

(2.14)
$\tilde{u}:=\frac{\delta^{1/p-1}\,{u}_{f}}{\|{u}_{f}\|_{L^{p^{*}}(\Omega)}+\|f\|_{L^{r}(\Omega)}}\qquad{\mbox{and}}\qquad\tilde{f}:=\frac{\delta^{1/p-1}\,f}{\|{u}_{f}\|_{L^{p^{*}}(\Omega)}+\|f\|_{L^{r}(\Omega)}},$

which satisfy

(2.15) $\begin{cases}\mathcal{L}_{s,p,\beta}\tilde{u}=\tilde{f}&\text{in
$\Omega$},\\\ \tilde{u}=0&\text{in
$\mathbb{R}^{n}\setminus\Omega$}.\end{cases}$

Now, for every $k\in\mathbb{N}$, we define the sequence of real numbers
$C_{k}:=1-2^{-k}$ and the auxiliary functions

$v_{k}:=\tilde{u}-C_{k},\quad w_{k}:=(v_{k})_{+}:=\max\\{v_{k},0\\},\quad
U_{k}:=\|w_{k}\|_{L^{p^{*}}(\Omega)}^{p}.$

We notice that, by the Dominated Convergence Theorem,

(2.16)
$\lim_{k\to+\infty}U_{k}=\lim_{k\to+\infty}\|w_{k}\|^{p}_{L^{p^{*}}(\Omega)}=\|(\tilde{u}-1)_{+}\|^{p}_{L^{p^{*}}(\Omega)}.$

Also, if we take $k:=0$, we see that
$w_{0}=(v_{0})_{+}=(\tilde{u}-C_{0})_{+}=\tilde{u}_{+}$. Now, denoting by
$p^{*}:=\frac{np}{n-p}$ the classical Sobolev critical exponent, we get that

(2.17)
$U_{0}=\left(\int_{\Omega}w_{0}^{p^{*}}(x)\,dx\right)^{p/p^{*}}\leq\left(\int_{\Omega}\tilde{u}^{p^{*}}(x)\,dx\right)^{p/p^{*}}=\|\tilde{u}\|^{p}_{L^{p^{*}}(\Omega)}\leq\delta^{p/p-1}.$

We will take it conveniently small in what follows. Now, since in
$\mathbb{R}^{n}\setminus\Omega$ we have that $v_{k+1}=-C_{k+1}\leq 0$ and thus

$w_{k+1}=0,$

we get that $w_{k+1}$ is an admissible test function, so that

(2.18)
$\begin{split}&\int_{\Omega}|\nabla\tilde{u}|^{p-2}\langle\nabla\tilde{u},\nabla
w_{k+1}\rangle\,dx+\iint_{\mathbb{R}^{2}n}\\!\\!\\!\\!\\!\frac{|\tilde{u}(x)-\tilde{u}(y)|^{p-2}(\tilde{u}(x)-\tilde{u}(y))(w_{k+1}(x)-w_{k+1}(y))}{|x-y|^{n+2s}}\,dx\,dy\\\
&\qquad=\int_{\Omega}w_{k+1}(x)\,\tilde{f}(x)\,dx.\end{split}$

We now exploit the fact that the nonlocal part has a positive sign to get rid
of it: indeed, for a.e. $x,y\in\mathbb{R}^{n}$, we have (see, e.g., [35, Lemma
10])

(2.19)
$\begin{split}|w_{k+1}(x)-w_{k+1}(y)|^{2}&=|(v_{k+1})_{+}(x)-(v_{k+1})_{+}(y)|^{2}\\\\[5.69046pt]
&\leq((v_{k+1})_{+}(x)-(v_{k+1})_{+}(y))(v_{k+1}(x)-v_{k+1}(y))\\\\[5.69046pt]
&=(w_{k+1}(x)-w_{k+1}(y))(\tilde{u}(x)-\tilde{u}(y)).\end{split}$

Moreover,

$\displaystyle\int_{\Omega}|\nabla\tilde{u}(x)|^{p-2}\langle\nabla\tilde{u}(x),\nabla
w_{k+1}(x)\rangle\,dx$
$\displaystyle=\int_{\Omega\cap\\{\tilde{u}>C_{k}\\}}|\nabla\tilde{u}(x)|^{p-2}\langle\nabla\tilde{u}(x),\nabla
v_{k+1}(x)\rangle\,dx$ $\displaystyle=\int_{\Omega}|\nabla
w_{k+1}(x)|^{p}\,dx.$

From this, (2.18) and (2.19) we can infer that

$\begin{split}&\int_{\Omega}|\nabla
w_{k+1}(x)|^{p}\,dx\leq\int_{\Omega}w_{k+1}(x)\,\tilde{f}(x)\,dx.\end{split}$

Hence, by Sobolev Inequality,

(2.20)
$U_{k+1}=\left(\int_{\Omega}|w_{k+1}(x)|^{p^{*}}\,dx\right)^{p/p^{*}}\leq
C\,\int_{\Omega}|\nabla w_{k+1}(x)|^{p}\,dx\leq
C\,\int_{\Omega}w_{k+1}(x)\,|\tilde{f}(x)|\,dx,$

for some $C>0$. Also, $v_{k+1}\leq v_{k}$ and therefore

(2.21) $w_{k+1}\leq w_{k}.$

Moreover, we observe that

$w_{k}=(\tilde{u}-C_{k})_{+}=\left(\tilde{u}-C_{k+1}+\frac{1}{2^{k+1}}\right)_{+}=\left(v_{k+1}+\frac{1}{2^{k+1}}\right)_{+},$

and, as a result,

(2.22)
$\\{w_{k+1}>0\\}=\\{v_{k+1}>0\\}\subseteq\left\\{w_{k}>\frac{1}{2^{k+1}}\right\\}.$

We now introduce the following number

(2.23)
$\tau:=p^{*}\,\left(p^{*}-\frac{p^{*}}{r}-1\right)^{-1}<\frac{p^{*}}{p-1},$

in such a way that

(2.24) $\frac{1}{p^{*}}+\frac{1}{r}+\frac{1}{\tau}=1.$

Thanks to the lower bound for $q$, we also have that

$p^{*}-\frac{p^{*}}{r}-1>p^{*}-\frac{p^{*}}{n/p}-1=\frac{np}{n-p}-\frac{p^{2}}{n-p}-1=p-1,$

and it clearly follows from its very definition that

$\tau>\frac{p^{*}}{p^{*}-1}>1.$

From this, (2.21) and (2.22), using Hölder inequality with exponents $p^{*}$,
$r$ and $\tau$, we get that

(2.25)
$\begin{split}&\int_{\Omega}w_{k+1}(x)\,|\tilde{f}(x)|\,dx=\int_{\Omega\cap\\{w_{k+1}>0\\}}w_{k+1}(x)\,|\tilde{f}(x)|\,dx\\\\[5.69046pt]
&\qquad\leq\|\tilde{f}\|_{L^{r}(\Omega)}\,\|w_{k+1}\|_{L^{p^{*}}(\Omega)}\,|\Omega\cap\\{w_{k+1}>0\\}|^{1/\tau}\\\\[5.69046pt]
&\qquad\leq\|w_{k}\|_{L^{p^{*}}(\Omega)}\,\left|\Omega\cap\left\\{w_{k}>\frac{1}{2^{k+1}}\right\\}\right|^{1/\tau}\\\\[5.69046pt]
&\qquad\leq
U_{k}^{1/p}\,\left(2^{p^{*}(k+1)}\int_{\Omega\cap\left\\{w_{k}>\frac{1}{2^{k+1}}\right\\}}w_{k}^{p^{*}}\right)^{1/\tau}\\\\[5.69046pt]
&\qquad\leq\tilde{C}^{k}\,U_{k}^{1/p}\,U_{k}^{p^{*}/(p\tau)},\end{split}$

where $\tilde{C}>1$. We now define

$\gamma:=\frac{1}{p}+\frac{p^{*}}{\tau p},$

and, thanks to (2.23), we can easily notice that

(2.26) $\gamma>1,$

By (2.20) and (2.25), we can notice that

$U_{k+1}\leq\hat{C}^{k}\;U_{k}^{\gamma},$

for some $\hat{C}>1$. As a result, recalling (2.17) and keeping in mind that
$\delta>0$ can be taken sufficiently small, we conclude that

$\lim_{k\to+\infty}U_{k}=0.$

This and (2.16) give that

$\text{$\|(\tilde{u}-1)_{+}\|^{p}_{L^{p^{*}}(\Omega)}=0$},$

and therefore $\tilde{u}\leq 1$. As a consequence, recalling (2.14), for every
$x\in\Omega$,

(2.27)
${u}_{f}(x)\leq\frac{\|{u}_{f}\|_{L^{p^{*}}(\Omega)}+\|f\|_{L^{r}(\Omega)}}{\delta},$

and this closes the proof. ∎

######  Corollary 2.5.

Assume that $\beta\in[0,1)$, $0<s<1$, $1<p<n$ and let $u$ be a weak solution
of (1.2). Then

(2.28) $u\in L^{\infty}(\mathbb{R}^{n})\quad\textrm{for every
}\alpha>\max\left\\{0,(q-1)\left(\dfrac{n}{p}-1\right)-p\right\\}.$

If $\beta=1$, $0<s<1$ is such that $1<sp<n$ and $u$ is a weak solution of
(1.2), then

(2.29) $u\in L^{\infty}(\mathbb{R}^{n})\quad\textrm{for every
}\alpha>\max\left\\{0,(q-1)\left(\dfrac{n}{p}-s\right)-sp\right\\}.$

###### Proof.

First we notice that if

(2.30) $r\left(\alpha+\left(1-\dfrac{n}{p}\right)(q-1)\right)+n>0,$

then

$|x|^{\alpha}u^{q-1}\in L^{r}(B).$

This follows noticing that, thanks to Lemma 2.3, it holds that

$\int_{B}|x|^{\alpha r}u^{r(q-1)}\,dx\leq C\int_{B}|x|^{\alpha
r}|x|^{(q-1)r(p-n)/p}\,dx.$

Now, it is clear that if

(2.31) $\alpha\geq\left(\dfrac{n}{p}-1\right)(q-1),$

then (2.30) is trivially satisfied for every $r>0$, in particular for
$r>\tfrac{n}{p}$. Therefore the conclusion follows from Theorem 2.4.
On the other hand, if

(2.32) $\alpha<\left(\dfrac{n}{p}-1\right)(q-1),$

it is enough to take

$r=\dfrac{-n}{(q-1)\left(1-\dfrac{n}{p}\right)+\alpha},$

and one can easily recognize that such $r>\tfrac{n}{p}$ if and only if

(2.33) $\alpha>\left(\dfrac{n}{p}-1\right)(q-1)-p.$

Combining (2.31), (2.32) and (2.33) we immediately get (2.28). An analogous
argument by using Lemma 2.2 and [8, Theorem 3.1] gives the result for
$\beta=1$.

∎

######  Remark 2.6.

It is clear that the maximum found in (2.28) in the case $\beta\in[0,1)$
imposes a new relation between $p$, $q$ and $n$. By quite simple computations
one can find bounds on $q$ in terms of $p$ and $n$ for which that maximum is
actually $0$. This happens e.g. if $n=2,3,4,5$ and $q<\tfrac{p^{2}+n-p}{n-p}$.
For $n\geq 6$ one has more restrictions on $p$. On the other hand, in the case
$\beta=1$, we find that $u\in L^{\infty}(\mathbb{R}^{n})$ for every $\alpha>0$
if we restrict the range of $0<s<\frac{n}{p}$ as

$0<s<\frac{n}{p}\frac{q-1}{p+q-1}.$

Finally, we recall the following well-known compactness result:

######  Lemma 2.7.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain, let $p>1$ and let
$\\{u_{n}\\}_{n\in\mathbb{N}}$ be a bounded sequence in $L^{p}(\Omega)$ such
that $\\{u_{n}\\}_{n\in\mathbb{N}}$ converges to $u$ a.e. Then $u\in
L^{p}(\Omega)$ and $u_{n}\to u$ in $L^{r}(\Omega)$ for $r\in[1,p)$.

##  3\. Existence and non-existence results

The proof of the existence result follows by using the mountain pass lemma
(Proposition A.2) due to Ambrosetti and Rabinowitz.

######  Lemma 3.1.

Let $p>1$ and $\alpha>0$. Then the following holds:

  * (i)

_(case $\beta=1$)_ if $s\in(0,1)$ and $n>sp$, then the inclusion
$W^{s,p}_{rad}(\mathbb{R}^{n})\subset L^{r}(|x|^{\alpha},B)$ is compact
provided that

(3.1) $1\leq r<\begin{cases}\frac{np}{n-sp-\alpha p}&\text{if
}\alpha<\frac{n-sp}{p},\\\\[3.0pt] \infty&\text{if
}\alpha\geq\frac{n-sp}{p}.\end{cases}$

  * (ii)

_(case $\beta\in[0,1)$)_ if $n\geq\max\\{2,p\\}$, then the inclusion
$W^{1,p}_{rad}(\mathbb{R}^{n})\subset L^{r}(|x|^{\alpha},B)$ is compact
provided that

(3.2) $1\leq r<\begin{cases}\frac{np}{n-p-\alpha p}&\text{if
}\alpha<\frac{n-p}{p},\\\\[3.0pt] \infty&\text{if
}\alpha\geq\frac{n-p}{p}.\end{cases}$

###### Proof.

We start proving (i), hence let $u\in W^{s,p}_{rad}(\mathbb{R}^{n})$. By
virtue of Lemma 2.2, we have

(3.3) $\displaystyle\begin{split}\int_{B}|x|^{\alpha r}|u|^{r}\,dx&\leq
c\|u\|_{W^{s,p}(\mathbb{R}^{n})}^{r}\int_{B}|x|^{\alpha
r+sr-\frac{rn}{p}}\,dx\\\
&=n\omega_{n}c\|u\|_{W^{s,p}(\mathbb{R}^{n})}^{r}\int_{0}^{1}\rho^{r(\alpha+s-\frac{n}{p})+n-1}\,d\rho\\\
&=n\omega_{n}c\|u\|_{W^{s,p}(\mathbb{R}^{n})}^{r}\frac{1}{r(\alpha+s-\frac{n}{p})+n}\end{split}$

provided that $n+r(\alpha+s-\frac{n}{p})>0$.

Now, let $\\{u_{n}\\}_{n\in\mathbb{N}}$ be a bounded sequence in
$W^{s,p}_{rad}(\mathbb{R}^{n})$. Up to a subsequence, by the compact embedding
theorem for fractional Sobolev spaces [14, Corollary 7.2],

$\displaystyle u_{n}\to u\text{ strongly in }L^{m}(B)\text{ for all
}m<p^{*}_{s}$ $\displaystyle u_{n}\to u\text{ a.e. in }B.$

Therefore,

$|x|^{\alpha}u_{n}\to|x|^{\alpha}u\text{ a.e. in }B.$

By estimate (3.3) the sequence $\\{|x|^{\alpha}u_{n}\\}_{n\in\mathbb{N}}$ is
bounded in $L^{r}(B)$ for all $r$ satisfying (3.1). In turn, in light of Lemma
2.7, it follows that

$|x|^{\alpha}u_{n}\to|x|^{\alpha}u\text{ strongly in }L^{q}(B)$

for all $q<r$, and then the lemma follows.
The proof of (ii) runs in an analogous way, so we omit the details. ∎

Let us check that $\mathcal{J}$ satisfies the Palais-Smale condition.

######  Lemma 3.2.

Let $p<q<p^{*}_{\beta,\alpha}$. For $\beta\in[0,1]$ the functional
$\mathcal{J}$ defined in (1.3) satisfies the conditions

  * (i)

$\mathcal{J}(0)=0$ and $\mathcal{J}(v)\leq 0$ for some $v\neq 0$ in
$X_{\beta,rad}(B)$,

  * (ii)

there exists $\mu\in\bigl{(}0,\|v\|_{\beta}\bigr{)}$ and $\sigma>0$ such that
$\mathcal{J}\geq\sigma$ on $S_{\mu}:=\\{u\in
X_{\beta,rad}(B)\,:\,\|u\|_{\beta}=\mu\\}$.

###### Proof.

Let us check (i). Obviously $\mathcal{J}(0)=0$. Let $u_{0}\in
X_{\beta,rad}(B)$ be a positive function such that $\|u_{0}\|_{\beta}=1$.
Then, since $q>p$, we have that there exists a positive constant $C>0$ such
that

(3.4)
$\displaystyle\mathcal{J}(tu_{0})\leq(1+C)\frac{t^{p}}{p}-\frac{t^{q}}{q}\int_{\Omega_{\rho}}|x|^{\alpha}(u^{+}_{0})^{q}\,dx\leq
0$

for $t>0$ large enough. The constant $C$ can be taken equal to zero in the
cases $\beta=0$ or $\beta=1$.

Let us check (ii). First, we consider the pure nonlcal case $\beta=1$. We
observe that $\mathcal{J}$ is well defined since for $x\in B$, by Lemma 2.2

(3.5) $\begin{split}\int_{B}|x|^{\alpha}(u^{+})^{q}\,dx&\leq
c[u]_{s,p}^{q}\int_{B}|x|^{\alpha+sq-\frac{nq}{p}}\,dx\\\
&=c[u]^{q}_{s,p}\int_{0}^{1}r^{\alpha+q(s-n/p)+n-1}\,dr\\\ &\leq
c[u]^{q}_{s,p}\end{split}$

for $q<p^{*}_{\beta,\alpha}$. Since $q>p$, from (3.5) we get that

(3.6)
$\mathcal{J}(u)\geq\frac{1}{p}[u]_{s,p}^{p}-c[u]_{s,p}^{q}=\frac{\mu^{p}}{p}-c\mu^{q}=\sigma>0$

if $\left[u\right]_{s,p}=\mu$ for $\mu>0$ small enough.
The case $\beta\in[0,1)$ works in a similar way replacing $[u]_{s,p}$ with
$\|\nabla u\|_{L^{p}(B)}$ and taking $s=1$. This closes the proof. ∎

######  Remark 3.3.

We note that the function $v$ in (i) is given by $tu_{0}$ in (3.4). Since $t$
can be taken large enough and $u_{0}$ has been normalized, we can infer that
$\|tu_{0}\|_{\beta}$ does not depend on $s$ and therefore the Mountain Pass
solution found in (ii) has a norm which is uniformly bounded (in $s$).

######  Lemma 3.4.

Let $p<q<p^{*}_{\beta,\alpha}$. Then, the functional $\mathcal{J}$ satisfies
the Palais-Smale condition for every $\beta\in[0,1]$

###### Proof.

Let us first prove that every Palais-Smale sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}\subset X_{\beta,rad}(B)$ for $\mathcal{J}$ is
bounded. Since $\mathcal{J}^{\prime}(u_{n})\to 0$,

(3.7) $\displaystyle|\langle\mathcal{J}^{\prime}(u_{n}),u_{n}\rangle|$
$\displaystyle=\left|(1-\beta)\|\nabla
u_{n}\|^{p}_{L^{p}(B)}+\beta\left[u_{n}\right]_{s,p}^{p}-\int_{B}|x|^{\alpha}(u_{n}^{+})^{q}\,dx\right|\leq\|u_{n}\|_{\beta},$

for $n$ large enough. The condition $|\mathcal{J}(u_{n})|\leq C$ is equivalent
to

(3.8) $\left|\frac{1-\beta}{p}\|\nabla
u_{n}\|^{p}_{L^{p}(B)}+\frac{\beta}{p}\left[u_{n}\right]_{s,p}^{p}-\frac{1}{q}\int_{B}|x|^{\alpha}(u_{n}^{+})^{q}\,dx\right|\leq
C.$

From (3.7) and (3.8) it follows that

$\displaystyle(1-\beta)\|\nabla u_{n}\|_{L^{p}(B)}^{p}+\beta[u_{n}]_{s,p}^{p}$
$\displaystyle\leq pC+\frac{p}{q}\int_{B}|x|^{\alpha}(u_{n}^{+})^{q}\,dx$
$\displaystyle\leq
pC+\frac{p}{q}\|u_{n}\|_{\beta}+\frac{(1-\beta)p}{q}\|\nabla
u_{n}\|_{L^{p}(\mathbb{R}^{n})}^{p}+\frac{\beta
p}{q}\left[u_{n}\right]_{s,p}^{p}$

from where, since $p<q$, we get that

$\left(1-\frac{p}{q}\right)((1-\beta)\|\nabla
u_{n}\|_{L^{p}(B)}^{p}+\beta[u_{n}]_{s,p}^{p})\leq
pC+\frac{p}{q}\|u_{n}\|_{\beta}.$

In particular, when $\beta=1$ this gives that

$\left(1-\frac{p}{q}\right)[u_{n}]_{s,p}^{p}\leq pC+\frac{p}{q}[u_{n}]_{s,p},$

and when $\beta\in[0,1)$,

$\left(1-\frac{p}{q}\right)\|\nabla u_{n}\|_{L^{p}(B)}^{p}\leq
pC+\frac{p}{q}\|\nabla u_{n}\|_{L^{p}(B)}.$

Hence, for $\beta\in[0,1]$ the sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ is
bounded in $X_{\beta,rad}(B)$. Passing now to a subsequence, we have
$u_{n}\rightharpoonup u$ in $X_{\beta,rad}(B)$. Let us first prove that

(3.9) $\lim_{n\to\infty}\int_{B}|x|^{\alpha}(u_{n}^{+})^{q-1}(u_{n}-u)\,dx=0.$

Up to a further subsequence, the sequence $\\{u_{n}-u\\}_{n\in\mathbb{N}}$
converges weakly to zero in $L^{p^{*}_{\beta}}(B)$. Then, in order to get
(3.9) we have to prove that $|x|^{\alpha}u_{n}^{q-1}\to|x|^{\alpha}u^{q-1}$
strongly in $L^{(p^{*}_{\beta})^{\prime}}$, where

(3.10)
$\left(p^{*}_{\beta}\right)^{\prime}=\left\\{\begin{array}[]{rl}\frac{np}{np+p-n}&\textrm{
if }\beta\in[0,1),\\\ \frac{np}{np+sp-n}&\textrm{ if
}\beta=1.\end{array}\right.$

This is equivalent to prove that

$|x|^{\alpha/(q-1)}u_{n}\to|x|^{\alpha/(q-1)}u\text{ strongly in
}L^{(q-1)(p^{\ast}_{\beta})^{\prime}}(B).$

By using Lemma 3.1, this is actually true whenever

(3.11)
$(q-1)(p^{\ast}_{\beta})^{\prime}<\left\\{\begin{array}[]{rl}\frac{np}{n-p-\frac{\alpha
p}{q-1}}&\textrm{ if }\beta\in[0,1),\\\ \frac{np}{n-sp-\frac{\alpha
p}{q-1}}&\textrm{ if }\beta=1,\end{array}\right.$

that is, when

$q<p^{*}_{\beta,\alpha}.$

Therefore, from (2.7), we have for all $n\in\mathbb{N}$,

$\displaystyle|\langle\mathcal{L}_{s,p,\beta}(u_{n}),u_{n}-u\rangle|$
$\displaystyle=\Bigl{|}\langle\mathcal{J}^{\prime}(u_{n}),u_{n}-u\rangle+\int_{B}|x|^{\alpha}(u_{n}^{+})^{q-1}(u_{n}-u)\,dx\Bigr{|}$
$\displaystyle\leq\|\mathcal{J}^{\prime}(u_{n})\|_{X^{*}_{\beta}(B)}\|u_{n}-u\|_{\beta}+\int_{B}|x|^{\alpha}(u_{n}^{+})^{q-1}(u_{n}-u)\,dx$

and the latter tends to $0$ as $n\to\infty$ by virtue of (3.9).

Finally, since by Remark 2.1, $\mathcal{L}_{s,p,\beta}$ fulfills the $({\bf
S})$-property of compactness, due to Proposition A.4 the previous computations
give that $u_{n}\to u$ strongly in $X_{\beta,rad}(B)$, which concludes the
proof. ∎

We are now in position to prove our existence result.

###### Proof of Theorem 1.1.

From Lemmas 3.2 and 3.4 we are in position to apply the Mountain pass Theorem
stated in Proposition A.2. Indeed, when $1<p<q<p^{*}_{\beta,\alpha}$ there is
a function $u\in X_{\beta,rad}(B)$ which is a critical point of $\mathcal{J}$
and hence is a non-trivial weak solution of (1.2). Moreover, Corollary 2.5 and
Remark 2.6 ensure that $u\in L^{\infty}(B)$ under our assumptions of the
parameters. ∎

As already mentioned, the non-existence result follows from [34].

###### Proof of Theorem 1.2.

Given $u\in X_{\beta}(B)$, denote $u_{\lambda}(x)=u(\lambda x)$ for
$\lambda>1$. For any $\beta\in[0,1]$ it is easy to check that

$\|u_{\lambda}\|_{\beta}\leq\lambda^{-\gamma}\|u\|_{\beta}$

with $\gamma=\frac{n-sp}{p}$ when $\beta\in[0,1)$ and $\gamma=\frac{n-p}{p}$
when $\beta=1$.

Denote $f(x,t)=|x|^{\alpha}t^{q-1}$ and $F(x,u)=\int_{0}^{u}f(x,t)\,dt$ and
let $u\in X_{\beta}(B)$ be a weak solution of (1.2). A straightforward
computation shows that $f$ is supercritical in the sense that

$\beta tf(x,t)>nF(x,t)+x\cdot F_{x}(x,t)\qquad\text{for all }t\in B\text{ and
}t\neq 0$

whenever $q>(n+\alpha)/\gamma$, i.e., when

(3.12) $q>\frac{p(n+\alpha)}{n-sp}\text{ when }\beta\in(0,1],\qquad
q>\frac{p(n+\alpha)}{n-p}\text{ when }\beta=1.$

If $u\in X_{\beta}(B)\cap W^{1,r}(B)\cap L^{\infty}(\Omega)$, by [34,
Proposition 1.4] we have that $u\equiv 0$.

In particular, if $u\in X_{\beta,rad}(B)$ is a weak solution of (1.2), by [14,
Proposition 2.2] we have that $u\in W^{1,p}(B)$; moreover, by Corollary 2.5
and Remark 2.6 $u\in L^{\infty}(B)$. As a consequence, by [34, Proposition
1.4] we obtain that $u\equiv 0$. ∎

##  4\. Stability of solutions

This section is devoted to prove our stability result for solutions of (1.2)
as $s\nearrow 1$.

We start with the following useful lemma.

######  Lemma 4.1.

Let $s_{k}\uparrow 1$ and $v_{k}\in W^{1,p}_{0}(B)$ be such that
$\sup_{k}\|\nabla v_{k}\|_{L^{p}(\mathbb{R}^{n})}^{p}<\infty$. Assume without
loss of generality that $v_{k}\to v$ strongly in $L^{p}(B)$. Then, for every
$u\in W^{1,p}_{0}(\Omega)$ we have that

$\langle\mathcal{L}_{s_{k},p,\beta}u,v_{k}\rangle\to\langle-\Delta_{p}u,v_{k}\rangle.$

###### Proof.

If we prove that for any $u\in W^{1,p}_{0}(B)$

(4.1)
$\langle-\Delta_{p}u,v\rangle\leq\liminf_{k\to\infty}\langle\mathcal{L}_{s_{k},p,\beta}u,v\rangle,$

then applying (4.1) to $-u$ gives the reverse inequality and hence the result.
By a refinement of [7, Section 3] we have

(4.2)
$\|\nabla(u+tv)\|_{L^{p}(\mathbb{R}^{n})}^{p}\leq\liminf_{k\to\infty}[u+tv_{k}]_{s_{k},p}^{p}$

Moreover, from [7] it follows that

(4.3) $\lim_{k\to\infty}[u]_{s_{k},p}^{p}=\|\nabla
u\|_{L^{p}(\mathbb{R}^{n})}^{p}.$

Denote $I:=\|\nabla(u+tv)\|_{L^{p}(\mathbb{R}^{n})}^{p}-\|\nabla
u\|_{L^{p}(\mathbb{R}^{n})}^{p}$. We write

$I=\beta I+(1-\beta)I.$

By using the lower semicontinuity of the $L^{p}$ norm, we have

$(1-\beta)I\leq(1-\beta)\liminf_{k\to\infty}\left(\|\nabla(u+tv_{k})\|_{L^{p}(\mathbb{R}^{n})}^{p}-\|\nabla
u\|_{L^{p}(\mathbb{R}^{n})}^{p}\right).$

From (4.2) and (4.3) we get

$\beta
I\leq\beta\liminf_{k\to\infty}\left([u+tv_{k}]_{s_{k},p}^{p}-[u]_{s_{k},p}^{p}\right).$

Then, from the last three relations and the superaditivity property of the
liminf we obtain

$I\leq\liminf_{k\to\infty}\left((1-\beta)\left(\|\nabla(u+tv_{k})\|_{L^{p}(\mathbb{R}^{n})}^{p}-\|\nabla
u\|_{L^{p}(\mathbb{R}^{n})}^{p}\right)+\beta\left([u+tv_{k}]_{s_{k},p}^{p}-[u]_{s_{k},p}^{p}\right)\right).$

Then, from [21, Lemma 2.7] it is immediate that

$\displaystyle\langle-\Delta_{p}u,v\rangle+o(1)$
$\displaystyle\leq\liminf_{k\to\infty}\left((1-\beta)\langle-\Delta_{p}u,v_{k}\rangle+\beta\langle(-\Delta_{p})^{s_{k}}u,v_{k}\rangle\right)+o(1)$
$\displaystyle=\liminf_{k\to\infty}\langle\mathcal{L}_{s_{k},p,\beta}u,v_{k}\rangle+o(1)$

from where (4.1) follows. ∎

###### Proof of Theorem 1.3.

The proof closely follows the one of [21, Theorem 3.3]. We start assuming that
$u_{k}\to u\quad\textrm{in }L^{p}(B)$. Since the sequence
$\\{u_{k}\\}_{k\in\mathbb{N}}$ is uniformly bounded in $X_{\beta,rad}(B)$, and
hence, due to [7], we can infer that $u\in W^{1,p}_{0}(B)$. Possibly passing
to a subsequence, we can also suppose that $u_{k}\to u$ a.e. in $B$.
Now, we define the sequence of functions $\\{\eta_{k}\\}_{k\in\mathbb{N}}$ as
follows:

(4.4) $\eta_{k}:=\mathcal{L}_{s_{k},p,\beta}u_{k}\in W^{-1,p^{\prime}}(B).$

By equation (1.2) and the fact that
$\sup_{k\in\mathbb{N}}\,\|u_{k}\|^{p}_{\beta}<\infty$, we get that the
sequence $\\{\eta_{k}\\}_{k\in\mathbb{N}}$ is actually bounded in
$W^{-1,p^{\prime}}(B)$, and therefore, possibly passing once again to a
subsequence, we can infer the existence of a function $\eta\in
W^{-1,p^{\prime}}(B)$ such that

(4.5) $\eta_{k}\rightharpoonup\eta\quad\textrm{weakly in
}W^{-1,p^{\prime}}(B).$

Now, since the $u_{k}$’s are weak solutions of (1.2), and exploiting the
appropriate convergences, we find that for every $v\in W^{1,p}_{0}(B)$ it
holds that

(4.6) $\displaystyle 0$
$\displaystyle=\langle\mathcal{L}_{s_{k},p,\beta},v\rangle-\int_{B}|x|^{\alpha}u_{k}^{q-1}v\,dx\to\langle\eta,v\rangle-\int_{B}|x|^{\alpha}u^{q-1}v\,dx\quad\textrm{as
}k\to+\infty.$

The monotonicity of both the $p$-Laplacian and the fractional $p$-Laplacian
implies that

(4.7) $\displaystyle 0$
$\displaystyle\leq\langle\mathcal{L}_{s_{k},p,\beta}u_{k},u_{k}-v\rangle-\langle\mathcal{L}_{s_{k},p,\beta}v,u_{k}-v\rangle$
$\displaystyle=\int_{B}|x|^{\alpha}u_{k}^{q-1}(u_{k}-v)\,dx-\langle\mathcal{L}_{s_{k},p,\beta}v,u_{k}-v\rangle.$

Passing to the limit as $k\to+\infty$, and using Lemma 4.1, we get

(4.8) $\displaystyle 0$
$\displaystyle\leq\int_{B}|x|^{\alpha}u^{q-1}(u-v)\,dx-\langle-\Delta_{p}v,u-v\rangle$
$\displaystyle=\langle\eta,u-v\rangle-\langle-\Delta_{p}v,u-v\rangle.$

Now, taking $v=u-tw$ for a given $w\in W^{1,p}_{0}(B)$ and with $t>0$, we
obtain that

(4.9) $\displaystyle 0$
$\displaystyle\leq\langle\eta,tw\rangle-\langle-\Delta_{p}(u-tw),tw\rangle$
$\displaystyle=\langle\eta,w\rangle-\langle-\Delta_{p}(u-tw),w\rangle$
$\displaystyle\to\langle\eta,w\rangle-\langle-\Delta_{p}u,w\rangle\quad\textrm{as
}t\to 0^{+}.$

This shows that $u\in W^{1,p}_{0}(B)$ is a weak solution of (1.2) with
$\beta=0$. ∎

##  5\. The linear case: existence of non-radial ground states

The aim of this section is, following the ideas of [38], to prove that under
suitable conditions on $\beta$ and $q$, there exists $\alpha^{*}>0$ such that,
ground states of $\mathcal{L}_{s,p,\beta}$ in the linear case are no radial
provided $\alpha>\alpha^{*}$.

We recall the following relation (see for instance [22, Lemma 4.2]).

######  Lemma 5.1.

Let $u\in W^{1,p}(\mathbb{R}^{n})$, $1\leq p<\infty$. Then, for each $0<s<1$

$\iint_{\mathbb{R}^{2n}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{n+2s}}\,dxdy\leq\frac{n\omega_{n}}{p}\left(\frac{1}{1-s}\|\nabla
u\|_{L^{p}(\Omega)}^{p}+\frac{2^{p}}{s}\|u\|_{L^{p}(\Omega)}^{p}\right).$

In particular, if $u\in X_{p}(B)$, for some $c=c(n,s,p)$

$\iint_{\mathbb{R}^{2n}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{n+2s}}\,dxdy\leq
c\|\nabla u\|_{p}.$

In order to simplify our notation, in this section we redefine the Gagiardo
seminorm as

(5.1)
$[u]_{s,p}:=\left(C(n,s,p)\iint_{\mathbb{R}^{2n}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{n+2s}}\,dx\,dy\right)^{1/p},$

where, in light of Lemma 5.1 the constant $C$ is chosen such that

(5.2) $[u]_{s,p}\leq\|\nabla u\|_{L^{p}(\Omega)}$

for all $u\in X_{p}(B)$.

Moreover, for $\beta\in[0,1)$ we set

$\mathcal{L}_{s,\beta}u:=(-\Delta)u+\beta(-\Delta)^{s}u.$

We say that $u\in X_{2}(B)$ is a _ground state_ of

$\mathcal{L}_{s,\beta}u=|x|^{\alpha}u^{q-1},\;u>0\text{ in }B$

if $u$ is a minimizer of

$\inf_{0\neq u\in X_{2}(B)}R(u)$

where

$R(u)=\frac{Z(u)}{N(u)}=\frac{\beta[u]_{s,2}^{2}+\|\nabla
u\|_{L^{2}(\Omega)}^{2}}{\left(\int_{B}|x|^{\alpha}|u|^{q}\,dx\right)^{\frac{2}{q}}}.$

######  Proposition 5.2.

Let $\alpha>0$, $0\leq\beta\leq 1$, $n\geq 3$ and $q>2+\beta$. Then any radial
minimizer of $R$ satisfies that

(5.3) $\beta[u]_{s,2}^{2}+\|\nabla
u\|_{L^{2}(B)}^{2}\leq\frac{(1+\beta)(n-1)}{q-2-\beta}\int_{B}\frac{u^{2}}{|x|^{2}}\,dx$

###### Proof.

We assume that $N(u)=1$. Let $u\in X_{2}(B)$ be a minimizer of $R$ and define
$g(t)=R(u+tv)$ for $t\geq 0$ and $v\in X_{2}(B)$. Then
$g^{\prime\prime}(0)\geq 0$ and $g^{\prime}(0)=0$, which gives

$g^{\prime\prime}(0)=\frac{1}{N^{2}(u)}\left(\langle
Z^{\prime\prime}(u)v,v\rangle N(u)-\langle N^{\prime\prime}(u)v,v\rangle
Z(u)\right)$

An easy computation shows that

$\langle Z^{\prime\prime}(u)v,v\rangle=2\left(\beta
C\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|v(x)-v(y)|^{2}}{|x-y|^{n+2s}}\,dxdy+\int_{B}|\nabla
v|^{2}\,dx\right)$

and that

$\langle
N^{\prime\prime}(u)v,v\rangle=2(2-q)\left(\int_{B}|x|^{\alpha}uv\,dx\right)^{2}+2(q-1)\int_{B}|x|^{\alpha}v^{2}\,dx.$

Then, $g^{\prime\prime}(0)\geq 0$ is equivalent to $I_{1}\leq I_{2}$ where

$\displaystyle
I_{1}:=\left((2-q)\left(\int_{B}|x|^{\alpha}uv\,dx\right)^{2}+(q-1)\int_{B}|x|^{\alpha}|uv|^{2}\,dx\right)(\beta[u]_{s,2}^{2}+\|\nabla
u\|_{L^{2}(B)}^{2})$ $\displaystyle I_{2}:=\left(\beta
C\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|v(x)-v(y)|^{2}}{|x-y|^{n+2s}}\,dxdy+\int_{B}|\nabla
v|^{2}\,dx\right)$

Observe that by (5.2)

$I_{2}\leq(1+\beta)\|\nabla v\|_{L^{2}(B)}^{2}.$

Assume $u$ is radial and choose $v$ of the form $v=u(r)f(\sigma)$, where $f$
is a smooth function defined in $S^{n-1}$ with zero mean. Observe that $v\in
X_{2}(B)$ for $n\geq 3$.

Since

$|\nabla v|^{2}=\left(\frac{\partial u}{\partial
r}\right)^{2}f^{2}+\frac{u^{2}}{r^{2}}|\nabla_{\sigma}f|^{2}$

we have that

$\displaystyle\frac{1}{1+\beta}I_{2}$ $\displaystyle\leq\int_{B}|\nabla
v|^{2}\,dx=\int_{B}|\nabla
u(|x|)|^{2}\,dx\int_{S^{n-1}}f^{2}\,dS_{\sigma}+\int_{B}\frac{u(|x|)^{2}}{|x|^{2}}\,dx\int_{S^{n-1}}|\nabla_{\sigma}f|^{2}\,dS_{\sigma}$
$\displaystyle\leq\left(\beta[u]_{s,2}^{2}+\|\nabla
u\|_{L^{2}(B)}^{2}\right)\int_{S^{n-1}}f^{2}\,dS_{\sigma}+\int_{B}\frac{u(|x|)^{2}}{|x|^{2}}\,dx\int_{S^{n-1}}|\nabla_{\sigma}f|^{2}\,dS_{\sigma};$

since $f$ has zero mean and $N(u)=1$ we get

$\displaystyle I_{1}$
$\displaystyle=\left((2-q)\left(\int_{B}|x|^{\alpha}u(|x|)\,dx\int_{S^{n-1}}f\,dS_{\sigma}\right)^{2}+(q-1)\int_{B}|x|^{\alpha}u(|x|)^{2}\,dx\int_{S^{n-1}}f^{2}\,dS_{\sigma}\right)Z(u)$
$\displaystyle=(q-1)(\beta[u]_{s,2}^{2}+\|\nabla
u\|_{L^{2}(B)}^{2})\int_{S^{n-1}}f^{2}\,dS_{\sigma}.$

Then $I_{1}\leq I_{2}$ means that

$(q-2-\beta)(\beta[u]_{s,2}^{2}+\|\nabla
u\|_{L^{2}(B)}^{2})\int_{S^{n-1}}f^{2}\,dS_{\sigma}\leq(1+\beta)\int_{B}\frac{u(|x|)^{2}}{|x|^{2}}\,dx\int_{S^{n-1}}|\nabla_{\sigma}f|^{2}\,dS_{\sigma}$

that is

$\frac{q-2-\beta}{1+\beta}(\beta[u]_{s,2}^{2}+\|\nabla
u\|_{L^{2}(B)}^{2})\leq\mathcal{S}_{n}(f)\int_{B}\frac{u^{2}}{|x|^{2}}\,dx$

where

$\mathcal{S}_{n}(f)=\frac{\int_{S^{n-1}}|\nabla_{\sigma}f|^{2}\,dS_{\sigma}}{\int_{S^{n-1}}f^{2}\,dS_{\sigma}}.$

Since the infimum of $\mathcal{S}_{n}(f)$ over all $f\in H^{1}(S^{n-1})$ with
$\int_{S^{n-1}}=0$ equals to $n-1$, we get

$\beta[u]_{s,2}^{2}+\|\nabla
u\|_{L^{2}(B)}^{2}\leq\frac{(1+\beta)(n-1)}{q-2-\beta}\int_{B}\frac{u^{2}}{|x|^{2}}\,dx$

and the proof concludes. ∎

We are now ready to prove Theorem 1.5.

###### Proof of Theorem 1.5.

Given $0<r\leq 1$, let $u_{\alpha}(r)\in X_{2}(B_{r})$ be the ground state of

$-\Delta u=|x|^{\alpha}u^{q-1},\;u>0\text{ in }B_{r}$

normalized such that $Z(u_{\alpha}(r))=1$. When $r=1$ we just write
$u_{\alpha}$.

_Step 1_. It holds that

(5.4) $\int_{B_{r}}|\nabla u_{\alpha}|^{2}\,dx\to 0\quad\text{ as
}\alpha\to\infty.$

Indeed, from the equation for $u_{\alpha}$, as in [38, Equation 16], we get

$\int_{B_{r}}|\nabla u_{\alpha}|^{2}\,dx\leq\int_{B_{r}}|\nabla
u_{\alpha}|^{2}\,dx+\beta[u_{\alpha}]_{s,2}^{2}\leq\frac{\int_{B_{r}}|x|^{\alpha}u_{\alpha}^{q}\,dx}{\int_{B}|x|^{\alpha}u_{\alpha}^{q}\,dx}.$

Now, since by (5.2),
$\|u_{\alpha}\|_{L^{2}(B_{r})}^{2}\leq\|u_{\alpha}\|_{L^{2}(B_{r})}^{2}+\beta[u_{\alpha}]_{s,2}^{2}\leq(1+\beta)\|u_{\alpha}\|_{L^{2}(B_{r})}^{2}$,
the claim follows similarly as in the proof of [38, Lemma 3.1].

_Step 2_. Consider the ground state $u_{\alpha}$ in $B$. Let us see that

(5.5) $\int_{B}\frac{u_{\alpha}^{2}}{|x|^{2}}\,dx\to 0\quad\text{ as
}\alpha\to\infty.$

Indeed, as in [38, Proposition 3.1, Step 1], observe that there exists $0<r<1$
independent of $\alpha$ such that $u_{\alpha}(r)<\varepsilon$.

We decompose $B$ as

$\displaystyle\int_{B}\frac{u_{\alpha}^{2}}{|x|^{2}}\,dx$
$\displaystyle=\int_{B_{r}}\frac{u_{\alpha}^{2}}{|x|^{2}}\,dx+\int_{A_{r}}\frac{u_{\alpha}^{2}}{|x|^{2}}\,dx$
$\displaystyle\leq
2\int_{B_{r}}\frac{\tilde{u}_{\alpha}^{2}}{|x|^{2}}\,dx+2\int_{B_{r}}\frac{u_{\alpha}(r)^{2}}{|x|^{2}}\,dx+\int_{A_{r}}\frac{u_{\alpha}^{2}}{|x|^{2}}\,dx$

where $A_{r}=B\setminus B_{r}$, $\tilde{u}_{\alpha}:=u_{\alpha}-u_{\alpha}(r)$
and $\tilde{u}_{\alpha}\in X_{2}(B_{r})$.

Observe that, as in [38, Equation 13], from Hardy’s inequality and (5.4) we
get

$\int_{B_{r}}\frac{\tilde{u}_{\alpha}^{2}}{|x|^{2}}\,dx\leq\frac{4}{(n-2)^{2}}\int_{B_{r}}|\nabla
u_{\alpha}|^{2}\,dx\to 0\quad\text{ as }\alpha\to\infty$

and, due to the election of $r$,

$\int_{B_{r}}\frac{u_{\alpha}(r)^{2}}{|x|^{2}}\,dx\leq
C\varepsilon^{2}\quad\text{ as }\alpha\to\infty.$

For the third term, observe that (5.4) gives in particular that

$u_{\alpha}(r)\rightharpoonup 0\quad\text{ weakly in }X_{2}\quad\text{ as
}\alpha\to\infty.$

which, together with Rellich-Kondrakov theorem gives

$\int_{A_{r}}\frac{u_{\alpha}^{2}}{|x|^{2}}\,dx\to 0\quad\text{ as
}\alpha\to\infty.$

Mixing up the last expressions we obtain (5.5).

_Step 3_. Finally, from (5.3) and (5.5) the result follows. ∎

##  Appendix A Mountain pass lemma

######  Definition A.1.

We say that the functional $\mathcal{J}$ satisfies the _Palais-Smale
compactness condition_ if each sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}\subset\mathcal{X}_{\rm rad}(B)$ such that

  * (i)

$\\{\mathcal{J}(u_{n})\\}_{n\in\mathbb{N}}$ is bounded, and

  * (ii)

$\mathcal{J}^{\prime}(u_{n})\to 0$ in $\mathcal{X}_{\rm rad}(B)$

is precompact in $\mathcal{X}_{\rm rad}(B)$.

We state the _mountain-pass theorem_ due to Ambrossetti and Rabinowitz [1].

######  Proposition A.2.

Let $E$ be a Banach space and let $\mathcal{J}\in C^{1}(E,\mathbb{R})$ satisfy
the Palais-Smale condition. Suppose that

  * (i)

$\mathcal{J}(0)=0$ and $\mathcal{J}(e)=0$ for some $e\neq 0$ in $E$;

  * (ii)

there exists $\rho\in(0,\|e\|)$, $\sigma>0$ such that $\mathcal{J}\geq\sigma$
in $S_{\rho}=\\{u\in E\colon\|u\|=\rho\\}$.

Then $\mathcal{J}$ has a positive critical value

$c=\inf_{h\in\Gamma}\max_{t\in[0,1]}\mathcal{J}(h(t))\geq\sigma>0$

where

$\Gamma=\\{h\in C([0,1],E)\colon h(0)=0,h(1)=e\\}.$

######  Definition A.3.

The functional $\mathcal{J}$ defined on $E$ satisfies the _$({\bf S})$
-property_ if $\\{u_{n}\\}_{n\in\mathbb{N}}$ is a sequence in $E$ such that
$u_{n}\rightharpoonup u$ weakly in $E$ and
$\langle\mathcal{J}(u_{n}),u_{n}-u\rangle\to 0$, then $u_{n}\to u$ strongly in
$E$.

The following result characterizes the _$({\bf S})$ -property_. See [33,
Proposition 1.3].

######  Proposition A.4.

Let $E$ be a uniformly convex Banach space and let $A_{p}\in
C^{1}(E,\mathbb{R})$ be such that

  * (i)

$\langle A_{p}(u),v\rangle\leq r\|u\|_{E}^{p-1}\|v\|_{E}$

  * (ii)

$\langle A_{p}(u),u\rangle=r\|u\|_{E}^{p}$

for some $r>0$, for all $u,v\in E$. Then $A_{p}$ satisfies the _$({\bf S})$
-property_.

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