# On the Eigenvalues of the $p\&q-$ Fractional Laplacian

Sabri Bahrouni  Mathematics Department, Faculty of Sciences, University of
Monastir, 5019 Monastir, Tunisia<EMAIL_ADDRESS>,  Hichem Hajaiej
Department of Mathematics, California State University at Los Angeles, Los
Angeles, CA 90032, USA<EMAIL_ADDRESS>and  Linjie Song  Institute of
Mathematics, AMSS, Academia Sinica, Beijing 100190, China  University of
Chinese Academy of Science, Beijing 100049, China
<EMAIL_ADDRESS>
###### Abstract.

We consider the eigenvalue problem for the fractional $p\&q-$Laplacian

(1) $\left\\{\begin{aligned}
(-\Delta)_{p}^{s}\,u+\mu(-\Delta)_{q}^{s}\,u+|u|^{p-2}u+\mu|u|^{q-2}u=\lambda\
V(x)|u|^{p-2}u\quad&\text{in }\Omega\\\
u=0\quad&\text{in}\quad\mathbb{R}^{N}\backslash\Omega,\end{aligned}\right.$

where $\Omega$ is an open bounded, and possibly disconnected domain,
$\lambda\in\mathbb{R}$, $1<q<p<\frac{N}{s}$, $\mu>0$ with a weight function in
$L^{\infty}(\Omega)$ that is allowed no change sign. We show that the problem
has a continuous spectrum. Moreover, our result reveals a discontinuity
property for the spectrum as the parameter $\mu\rightarrow 0^{+}.$ In
addition, a stability property of eigenvalues as $s\rightarrow 1^{-}$ is
established.

###### Key words and phrases:

fractional $p\&q-$Laplacien; eigenvalues; continuous spectrum; stability of
eigenvalues.
2010 Mathematics Subject Classifications: 35P30, 47J10.

##  1\. Introduction

In this paper we study the following eigenvalue problem for the Dirichlet
fractional $p\&q-$Laplacian

(2) $\left\\{\begin{aligned}
(-\Delta)_{p}^{s}\,u+\mu(-\Delta)_{q}^{s}\,u+|u|^{p-2}u+\mu|u|^{q-2}u=\lambda\
V(x)|u|^{p-2}u\quad&\text{in }\Omega\\\
u=0\quad&\text{in}\quad\mathbb{R}^{N}\backslash\Omega,\end{aligned}\right.$

where $\Omega\subset\mathbb{R}^{N}$ is a bounded domain,
$\lambda\in\mathbb{R}$, $1<q<p<\frac{N}{s}$, $\mu>0$ and $V\in
L^{\infty}(\Omega)$ is such that the Lebesgue measure of $\\{x\in\Omega\colon\
V(x)>0\\}$ is positive. The operator $(-\Delta)_{\alpha}^{s}$, with
$\alpha\in\\{p,q\\}$, is the so called fractional $\alpha$-Laplacian operator
which, up to normalization factors, may be defined for every function
$u\in\mathcal{C}_{c}^{\infty}\left(\mathbb{R}^{N}\right)$ as

$(-\Delta)_{\alpha}^{s}u(x)=2\lim_{r\rightarrow
0}\int_{\mathbb{R}^{N}\backslash\mathcal{B}_{r}(x)}\frac{|u(x)-u(y)|^{\alpha-2}(u(x)-u(y))}{|x-y|^{N+s\alpha}}dy\quad\left(x\in\mathbb{R}^{N}\right).$

Problems of this type appear in the case of two different materials that
involve power hardening exponents $p$ and $q$. Let us point out that the
combination of the operators $(-\Delta)_{p}^{s}\,u$ and $(-\Delta)_{q}^{s}$
appearing in problem (2) can be seen as a class of fractional unbalanced
double phase problems. Such patterns are strictly connected with mathematical
physics (fractional white-noise limit, fractional quantum mechanics,
fractional superdiffusion, etc), see [3]. This operator also can be regarded
as the fractional analog of the $p\&q-$Laplacian $-\Delta_{p}-\Delta_{q}$,
which comes from the study of general reaction diffusion equations with
nonhomogeneous diffusion

$u_{t}=\operatorname{div}[A(u)\nabla u]+c(x,u)\text{ and }A(u)=|\nabla
u|^{p-2}+|\nabla u|^{q-2}.$

These kinds of problems stem from a wide range of important applications,
including models of elementary particles, plasma physics, biophysics,
reaction-diffusion equations, elasticity theory, etc, see the survey paper
[39] for a more detailed account. Several results for $p\&q-$Laplacian
problems obtained in bounded domains and in $\mathbb{R}^{N}$ can be found in
[11, 13, 14, 25, 32, 45] and the references therein. Due to the numerous
applications and challenging problems due to the nature of the fractional
$p\&q-$ operators, the corresponding non-local problems have attracted the
attention of many researchers in recent years. For instance, see [1, 2, 3, 4,
10] for the existence and multiplicity results for problems involving the
fractional $p\&q-$Laplacian. However, to the best of our knowledge, there is
not much literature available regarding eigenvalues problems including the
fractional $p\&q-$Laplacian. The latter is crucial to understand and to derive
many critical properties of the operator.

The study of eigenvalue problems has been a challenging labor whose beginning
dates back to the mid-20th century, more precisely: Given two functionals
$\mathcal{A}$ and $\mathcal{B}$ defined on a suitable space $\mathcal{X}$ and
a prescribed number $c$, the task of analyzing the existence of numbers
$\lambda\in\mathbb{R}$ and elements $u\in\mathcal{X}$ satisfying (in some
appropriated sense) equations of the type

$\mathcal{B}^{\prime}(u)=\lambda\mathcal{A}^{\prime}(u),\qquad\mathcal{A}(u)=c,$

where $\mathcal{A}^{\prime}$ and $\mathcal{B}^{\prime}$ denote the Fréchet
derivatives of the functionals.

The prototypical $p-$Laplace operator ($p>1$) then became a focus of study,
and in particular, to understand its spectral structure: Given an open and
bounded set $\Omega\subset\mathbb{R}^{N}$, to determine the existence couples
$(\lambda,u)$ satisfying the equation

(3) $-div(|\nabla u|^{p-2}u)=\lambda|u|^{p-2}u\text{ in }\Omega,\quad
u=0\text{ on }\partial\Omega$

in a suitable sense. In the seminal work of García Azorero and Peral Alonso
[34], they proved the existence of a variational sequence of eigenvalues
tending to $+\infty$. However, they couldn’t establish whether it exhausts the
spectrum unless $p=2$ or $N=1$. Several properties on eigenvalues (and their
corresponding eigenfunctions) were addressed by Anane et al [5, 6] and
Lindqvist [37], among others, and also for more general boundary conditions
than Dirichlet.

The non-local counterpart of problem (3) governed by the well-known
_fractional $p-$Laplace_ operator takes the form

(4) $(-\Delta_{p})^{s}u=\lambda|u|^{p-2}u\text{ in }\Omega,\quad u=0\text{ in
}\mathbb{R}^{N}\setminus\Omega.$

The main difficulty here comes from the fact that this operator takes into
account long-range interactions. Problem (4) was introduced in [36]. Several
properties on eigenvalues and eigenfunctions were obtained in [27, 33, 36].
Allowing a growth behavior more general than power in problem (4) existence of
eigenvalues was studied in [8]. See also [9, 24, 38] for problems involving
the mixed fractional Laplacians.

Letting $\mu\rightarrow 0^{+}$, equation (2) becomes

(5) $\left\\{\begin{aligned} (-\Delta)_{p}^{s}\,u+|u|^{p-2}u=\lambda\
V(x)|u|^{p-2}u\quad&\text{in }\Omega\\\
u=0\quad&\text{in}\quad\mathbb{R}^{N}\backslash\Omega,\end{aligned}\right.$

We define

(6) $\displaystyle\lambda_{1}(p,V)=\inf_{u\in W^{s,p}_{0}(\Omega),\atop u\neq
0}\left\\{\frac{[u]_{s,p}^{p}+\|u\|_{p}^{p}}{\int_{\Omega}V(x)|u|^{p}dx}\colon\quad\int_{\Omega}V(x)|u|^{p}dx>0\right\\},$

where $[\cdot]_{s,p}$ is the Gagliardo semi-norm (see Section 2 for the
precise definition).

By Theorem 3.3 of [40] for the special linear case of the fractional Laplacian
and $V\equiv 1$, the infimum in (6) is achieved and $\lambda_{1}(p,1)>0$.

We aim to give a complete study of the spectrum of the previous equation and
we plan to prove the following statements:

  1. (i)

There exists a nondecreasing sequence of nonnegative eigenvalues obtained by
the Alexander-Spanier Cohmology with $\mathbb{Z}_{2}-$coefficients method
$(\lambda_{k})$ tending to $\infty$ as $k\rightarrow\infty$.

  2. (ii)

The first eigenvalue $\lambda_{1}(p,V)$ is simple and only eigenfunctions
associated with $\lambda_{1}(p,V)$ do not change sign.

  3. (iii)

The first eigenvalue $\lambda_{1}(p,V)$ is isolated.

  4. (iv)

We characterize the second eigenvalue by means of variational formulations,
where the second eigenvalue $\lambda_{2}(p,V)$ is well-defined, i.e.,

$\lambda_{2}(p,V)=\min\\{\lambda\colon\ \lambda\ \text{is an eigenvalue of
\eqref{quation-p} and}\ \lambda>\lambda_{1}(p,V)\\}>\lambda_{1}(p,V).$

Because of the non-homogeneity of the operator in problem (2), many of the
properties that eigenvalues of problem (5) are not inherited in the non-
homogeneous case: for instance, isolation, simplicity and a variational
characterization of the first eigenvalue, or a variational formula for the
second one. Moreover, the spectrum of (2) will be continuous, and in
principle, it is not clear the meaning of a first or second eigenvalue. In
other word if we denote by $\sigma^{s}_{p,q}(V)$ the spectrum of (2), which
main

$\sigma^{s}_{p,q}(V)=\left\\{\lambda\in\mathbb{R}\colon\ \lambda\ \text{is an
eigenvalue of}\ \eqref{mainequation}\right\\},$

we can completely describe the spectrum as follows:

$\sigma^{s}_{p,q}(V)=\begin{cases}(\lambda_{1}(p,V),+\infty),&\mbox{if }V\geq
0\\\
(-\infty,-\lambda_{1}(p,-V))\cup(\lambda_{1}(p,V),+\infty),&\mbox{otherwise}.\end{cases}$

The main result reads:

######  Theorem 1.1.

  1. $(i)$

If $-\lambda_{1}\left(p,-V\right)\leq\lambda\leq\lambda_{1}\left(p,V\right)$
holds, then for any $\mu>0$, (2) has no non-trivial solutions.

  2. $(ii)$

If $\lambda_{1}\left(p,V\right)\leq\lambda\leq\lambda_{2}\left(p,V\right)$
holds, then for any $\mu>0$, (2) has no sign-changing solutions.

  3. $(iii)$

Tf $\lambda>\lambda_{1}\left(p,V\right)$ or
$\lambda<-\lambda_{1}\left(p,-V\right)$ holds, then for any $\mu>0$, (2) has
at least one positive solution.

Our second aim, is to analyze the stability of eigenvalues for equation (2)
when the fractional parameter goes to $1^{-}$, in which case the limiting
problem of (2) is formally given by

(7) $\left\\{\begin{aligned}
(-\Delta)_{p}\,u+\mu(-\Delta)_{q}\,u+|u|^{p-2}u+\mu|u|^{q-2}u=\lambda\
V(x)|u|^{p-2}u\quad&\text{in }\Omega\\\
u=0\quad&\text{on}\quad\partial\Omega.\end{aligned}\right.$

More precisely, we aim to show that any accumulation point $\lambda$ of the
set $\left\\{\lambda^{s}\right\\}_{s\in(0,1)}$ (different from
$\lambda_{1}(p,V)$) belongs to $\sigma_{p,q}^{1}$, where
$\lambda^{s}\in\sigma_{p,q}^{s}$ is such that
$\sup_{s\in(0,1)}\lambda^{s}<\infty$ (see Theorem 5.2). For the limit problem
(7), the continuity of the spectrum has been first studied by [41]. This
investigation pursued in the second part of the paper is based upon the
results by Bourgain, Brezis, and Mironescu [17, 18]. We have that if $u\in
W_{0}^{1,p}(\Omega)$

$\lim_{s\nearrow 1}(1-s)[u]_{s,p,\mathbb{R}^{N}}^{p}=K(p,N)\|\nabla
u\|_{p,\Omega}^{p},$

where the constant $K(p,N)$ is given by

$K(p,N):=\frac{1}{p}\int_{\mathbb{S}^{N-1}}|\langle\sigma,\mathbf{e}\rangle|^{p}d\mathcal{H}^{N-1}(\sigma),\quad\mathbf{e}\in\mathbb{S}^{N-1}.$

This line of research was developed, for instance, in [15, 16, 22, 23, 30,
31], and was found to be applicable to many cases.

The rest of this paper is organized as follows. Section 2 includes
preliminaries and useful results about fractional order Sobolev spaces, the
fractional $p-$Laplacian, and the fractional $p\&q-$Laplacian. In Section 3,
we give a complete study of the spectrum when $\mu=0$. Then we study the case
when $\mu>0$ in Section 4 and the case when $s\rightarrow 1^{-}$ in Section 5.

##  2\. Preliminaries and useful results

###  2.1. Fractional order Sobolev spaces

Given a fractional parameter $0<s<1$ and $1\leq p<\infty$, we define the
fractional order Sobolev space $W^{s,p}\left(\mathbb{R}^{N}\right)$ as

$W^{s,p}\left(\mathbb{R}^{N}\right):=\left\\{u\in
L^{p}\left(\mathbb{R}^{N}\right):[u]_{s,p}<\infty\right\\}$

where the Gagliardo semi-norm $[\cdot]_{s,p}$ is defined as

$[u]_{s,p}^{p}:=K(N,s,p)\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dxdy.$

The constant $K(N,s,p)$ is a normalizing constant defined as

$K(N,s,p):=(1-s)\mathcal{K}(N,p),\qquad\text{ where
}\quad\mathcal{K}(N,p)^{-1}:=\frac{1}{p}\int_{\mathbb{S}^{N-1}}\omega_{n}^{p}\,dS_{\omega}$

and $\mathbb{S}^{N-1}$ is the unit sphere in $\mathbb{R}^{N}$.

The main feature of this constant is that, for any $u\in
L^{p}(\mathbb{R}^{N})$, it holds that

$\lim_{s\uparrow 1}[u]_{s,p}^{p}=\|\nabla u\|_{p}^{p},$

where the limit above is understood as an equality if $u\in
W^{1,p}(\mathbb{R}^{N})$ and $\liminf_{s\uparrow 1}[u]_{s,p}^{p}=\infty$
otherwise. This is the celebrated result obtained by Bourgain, Brezis and
Mironescu in [17].

The space $W^{s,p}(\mathbb{R}^{N})$ is a Banach space with norm given by
$\|u\|_{s,p}=\left(\|u\|_{p}^{p}+[u]_{s,p}^{p}\right)^{\frac{1}{p}}$. This is
a separable space and reflexive for $p>1$. Given an open set
$\Omega\subset\mathbb{R}^{N}$, we then consider

$W_{0}^{s,p}(\Omega):=\left\\{u\in
W^{s,p}\left(\mathbb{R}^{N}\right):u=0\text{ a.e. in
}\mathbb{R}^{N}\backslash\Omega\right\\}.$

By Poincaré inequality, it follows that $[\cdot]_{s,p}$ defines a norm in
$W_{0}^{s,p}(\Omega)$ equivalent to $\|\cdot\|_{s,p}.$

It is well-known that $W_{0}^{s,p}(\Omega)\hookrightarrow L^{r}(\Omega)$ with

$\begin{array}[]{ll}r\in(1,p^{*}_{s})&\quad\text{ if }sp<N\\\
r=p_{s}^{*}&\quad\text{ if }sp\neq N\\\ r\in(1,\infty)&\quad\text{ if
}sp=N\end{array}$

We also consider the usual $W_{0}^{1,p}(\Omega)$ endowed with the norm
$[u]_{1,p}^{p}=\int_{\Omega}|\nabla u|^{p}dx$.

###  2.2. The fractional $p-$Laplacian

For any $0<s\leq 1\leq p<\infty$ we consider the energy functional

$\displaystyle\mathcal{J}_{s,p}\colon
W^{s,p}_{0}(\Omega)\rightarrow\mathbb{R},\qquad\mathcal{J}_{s,p}(u)=\begin{cases}\frac{1}{p}[u]_{s,p}^{p}&\text{
if }0<s<1\\\ \frac{1}{p}\|\nabla u\|_{p}^{p}&\text{ if }s=1.\end{cases}$

We define a nonlinear operator $A_{s}\colon W^{s,p}_{0}(\Omega)\rightarrow
W^{-s,p^{\prime}}(\Omega)$

$\displaystyle\left\langle A_{s}(u),v\right\rangle$
$\displaystyle=\frac{K(N,s,p)}{2}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+sp}}\,dxdy$
$\displaystyle\left\langle A_{1}(u),v\right\rangle$
$\displaystyle=\int_{\Omega}|\nabla u|^{p-2}u\cdot\nabla v\,dx.$

The functional $\mathcal{J}_{s,p}$ is Fréchet differentiable and
$\mathcal{J}^{\prime}_{s,p}\colon W^{s,p}_{0}(\Omega)\rightarrow
W^{-s,p^{\prime}}(\Omega)$ is continuous and for every $u,v\in
W^{s,p}_{0}(\Omega)$

$\displaystyle\left\langle\mathcal{J}_{s,p}^{\prime}(u),v\right\rangle$
$\displaystyle=\left\langle A_{s}(u),v\right\rangle$
$\displaystyle\left\langle\mathcal{J}_{1,p}^{\prime}(u),v\right\rangle$
$\displaystyle=\left\langle A_{1}(u),v\right\rangle.$

Therefore, for any $0<s\leq 1<p$ we define the _fractional $p-$Laplace
operator_ as $(-\Delta_{p})^{s}:=\mathcal{J}^{\prime}_{s,p}$.

We recall a useful property of the fractional $p-$Laplacian. Since
$W^{s,p}_{0}(\Omega)$ is uniformly convex, it follows from [42, Proposition
1.3] that the operator $(-\Delta_{p})^{s}$ is of type $(S)$, i.e., every
sequence $(u_{j})\subset W^{s,p}_{0}(\Omega)$ such that $u_{j}\rightarrow u$
weakly and $(\mathcal{J}^{\prime}_{s,p}(u_{j}),u_{j}-u)\rightarrow 0$ has a
subsequence that converges strongly to $u$.

The usual $p-$Laplace operator can be seen as the limit of the fractional
$p-$Laplacian as $s\uparrow 1$ in the following sense:

######  Lemma 2.1.

[30, Lemma 2.8] Let $s_{k}\uparrow 1$ and $v_{k}\in W_{0}^{s,p}(\Omega)$ be
such that $\sup_{k\in\mathbb{N}}\left[v_{k}\right]_{s_{k},p}^{p}<\infty$.
Assume, without loss of generality, that $v_{k}\rightarrow v$ strongly in
$L^{p}(\Omega).$ Then, for every $u\in W_{0}^{1,p}(\Omega)$, we have

$\left\langle\left(-\Delta_{p}\right)^{s_{k}}u,v_{k}\right\rangle\rightarrow\left\langle-\Delta_{p}u,v\right\rangle.$

###  2.3. Setting of the problem

In the case of the usual Sobolev spaces, for any $1\leq p<q\leq\infty$, it is
easy to see that $W^{1,q}_{0}(\Omega)\subset W^{1,p}_{0}(\Omega)$. In the
fractional case this kind of embedding is NOT TRUE. In fact, in [21, Lemma
2.6] it is proved that

$W^{s_{1},p}_{0}(\Omega)\hookrightarrow W^{s_{2},q}_{0}(\Omega)\quad\text{for
any }0<s_{1}<s_{2}<1\leq q<p<\infty,$

this also holds when $p=q$ (see [35, Theorem 2.2]). However, the embedding

$W^{s,p}_{0}(\Omega)\hookrightarrow W^{s,q}_{0}(\Omega)\quad\text{ for any
}0<s<1\leq q<p<\infty.$

is not true (see [43, Theorem 1.1]). So, in order to deal with our problem
(2), we consider the space

$\mathcal{W}^{s}:=W^{s,p}_{0}(\Omega)\cap W^{s,q}_{0}(\Omega)$

endowed with the norm $[\cdot]_{s}:=[\cdot]_{s,p}+[\cdot]_{s,q}$.

We also consider the energy functional related to equation (2) given by

$\mathcal{J}_{s}(u)=\frac{1}{p}[u]_{s,p}^{p}+\frac{\mu}{q}[u]_{s,q}^{q}.$

It is well-defined on the space $\mathcal{W}^{s}$ for $0<s\leq 1$. In fact, it
is straightforward to see that $\mathcal{J}_{s}\in
C^{1}\left(\mathcal{W}^{s},\mathbb{R}\right)$ and its Fréchet derivative is
given by

$\displaystyle\left\langle\mathcal{J}_{s}^{\prime}(u),v\right\rangle=\left\langle\mathcal{J}_{s,p}^{\prime}(u),v\right\rangle+\mu\left\langle\mathcal{J}_{s,q}^{\prime}(u),v\right\rangle$

for any $u,v\in\mathcal{W}^{s}$.

The expression $\mathcal{J}^{\prime}_{s}$ defines what we call _fractional
$p\&q-$Laplacian_.

We say that a function $u\in\mathcal{W}^{s}$ is an eigenfunction of (2)
corresponding to the eigenvalue $\lambda$ if

$\left\langle\mathcal{J}_{s}^{\prime}(u),v\right\rangle+\int_{\Omega}(|u|^{p-2}u+\mu|u|^{q-2}u)v\,dx=\lambda\int_{\Omega}V(x)|u|^{p-2}uv\,dx\quad\text{
for all }v\in\mathcal{W}^{s}.$

###  2.4. Some properties of the fractional $p\&q-$Laplacian

It is well known that the operator $(-\Delta_{p})^{s}$ is strictly monotone.
This is a consequence of the well known Simon inequality (see [44])

$\left(|a|^{p-2}a-|b|^{p-2}b\right)\cdot(a-b)\geq\begin{cases}c|a-b|^{p}&\text{
if }p\geq 2\\\ c\frac{|a-b|^{2}}{(|a|+|b|)^{2-p}}&\text{ if
}1<p<2,\end{cases}$

for any $a,b\in\mathbb{R}^{N}(N\in\mathbb{N})$, where the constant $c$ depends
on $p$ and $N$. From the monotonicity of $(-\Delta_{p})^{s}$, it is easy to
see that

$\langle\mathcal{J}^{\prime}_{s}(u)-\mathcal{J}^{\prime}_{s}(v),u-v\rangle\geq
0\qquad u,v\in\mathcal{W}^{s}.$

Moreover, from Lemma 2.1 it is immediate to see that
$\mathcal{J}^{\prime}_{s}(u)\rightarrow\mathcal{J}^{\prime}_{1}(u)$ as
$s\uparrow 1$ in the following sense:

######  Lemma 2.2.

Let $s_{k}\uparrow 1$ and $v_{k}\in\mathcal{W}^{s_{k}}$ be such that
$\sup_{k\in\mathbb{N}}\left[v_{k}\right]_{s_{k}}<\infty$. Assume, without loss
of generality, that $v_{k}\rightarrow v$ strongly in $L^{p}(\Omega).$ Then,
for every $u\in W_{0}^{1,p}(\Omega)$, we have

$\langle\mathcal{J}^{\prime}_{s_{k}}(u),v_{k}\rangle\rightarrow\langle\mathcal{J}^{\prime}_{1}(u),v\rangle,\quad\text{for
any}\quad\mu>0.$

The previous lemma says that, for any $\mu>0$,

$\left\langle\left(-\Delta_{p}\right)^{s_{k}}u+\mu\left(-\Delta_{q}\right)^{s_{k}}u,v_{k}\right\rangle\rightarrow\left\langle-\Delta_{p}u-\mu\Delta_{q}u,v\right\rangle.$

##  3\. The case $\mu\rightarrow 0^{+}$

###  3.1. The first eigenvalue

In this section we assume that the potential $V$ is a positive weight of class
$L^{r}(\Omega)$, with $r>\frac{N}{sp}$. Let

$L^{p}(\Omega,V)=\left\\{u:\Omega\rightarrow\mathbb{R}\text{ measurable
}:V|u|^{p}\in L^{1}(\Omega)\right\\}.$

This space is endowed with the norm $\|\cdot\|_{p,V}$ which is defined by

$\|u\|_{p,V}^{p}=\int_{\Omega}V(x)|u(x)|^{p}\,dx.$

We remark that the embedding $W_{0}^{s,p}(\Omega)\hookrightarrow
L^{p}(\Omega,V)$ is compact. Indeed, by our assumption, for any $p>1$,
$r^{\prime}p<p^{*}$, and then, by [28, Corollary 7.2]
$W_{0}^{s,p}(\Omega)\hookrightarrow L^{r^{\prime}p}(\Omega)$ is compact, and
$L^{r^{\prime}p}(\Omega)\hookrightarrow L^{p}(\Omega,V)$ is continuous, since
$\|u\|_{p,V}^{p}\leq\|V\|_{r}\|u\|_{r^{\prime}p}^{p}$ for all $u\in
L^{r^{\prime}p}(\Omega)$ by the Hölder inequality.

######  Proposition 3.1.

The infimum in (6) is attained by a function $u_{1}\in W^{s,p}_{0}(\Omega)$
normalized such that $\int_{\Omega}V(x)|u|^{p}dx=1$.

###### Proof.

Observe that we can write

$\lambda_{1}(p,V)=\inf_{u\in\mathcal{S}}\frac{\mathcal{P}(u)}{\mathcal{B}(u)}$

with $\mathcal{P}(u)=[u]_{s,p}^{p}+\|u\|_{p}^{p}$,
$\mathcal{B}(u)=\|u\|_{p,V}^{p}$ and $\mathcal{S}=\\{u\in
W^{s,p}_{0}(\Omega)\backslash\\{0\\},\ \|u\|_{p,V}=1\\}$. Then, since
$\mathfrak{B}$ is continuous with respect to weak sequential convergence,
$\mathcal{B}^{{}^{\prime}}(u)=0$ only at $u=0$, and $\mathcal{P}$ is coercive
on $W^{s,p}_{0}(\Omega)\cap\\{\mathfrak{B}(u)\leq c\\}$, we can apply, for
instance, Theorem 6.3.2 of [12] to conclude the result. ∎

######  Proposition 3.2 (Minimum principle).

If $u$ is a nonnegative eigenfunction of (5) associated to $\lambda$ then
$u>0$ in $\Omega$.

###### Proof.

In light of [20, Theorem A.1] it is enough to prove that if $u$ is an
eigenfunction associated to $\lambda$ then $u\neq 0$ in all connected
components of $\Omega$. Suppose, by contradiction, that there is $K$ a
connected component of $\Omega$ such that $u\equiv 0$ in $K$.

Taking $\psi\in C^{\infty}_{c}(K)$ as a test function, we get

$\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\psi(x)-\psi(y))}{|x-y|^{N+sp}}dxdy=0$
$\displaystyle\Leftrightarrow$
$\displaystyle\int_{K}\psi(y)\int_{\Omega\backslash
K}(u(x))^{p-1}\left(\frac{1}{|x-y|^{N+sp}}\right)dxdy=0,\ \forall\psi\in
C^{\infty}_{c}(K).$

Then $u=0$ in $\Omega$, which is a contradiction. ∎

We prove next lower bound for the Lebesgue measure of $\Omega_{\pm}$.

######  Lemma 3.1.

If $u$ is an eigenfunction associated to $\lambda>\lambda_{1}(p,V)$, then

$(\lambda
C\|V\|_{r})^{\frac{1}{\frac{p}{\alpha}-\frac{1}{r^{\prime}}}}\leq|\Omega_{\pm}|$

where $\alpha\in\left(pr^{\prime},p_{s}^{*}\right)$ and is a constant
independent of $V,\lambda$ and $u$.

###### Proof.

Due to Proposition 3.3, $u_{\pm}$ are not trivial. Observe that $u_{+}\in
W^{s,p}(\Omega)$ and

$\left|u_{+}(x)-u_{+}(y)\right|^{p}\leq|u(x)-u(y)|^{p-2}(u(x)-u(y))\left(u_{+}(x)-u_{+}(y)\right)$

for all $(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$. Then, using Hölder’s
inequality, we have

$\displaystyle[u_{+}]_{s,p}^{p}$
$\displaystyle\leq\langle(-\Delta_{p})^{s}u,u_{+}\rangle=\lambda\int_{\Omega}V(x)u_{+}^{p}\,dx-\int_{\Omega}u_{+}^{p}\,dx$
$\displaystyle\leq\lambda\|V\|_{r}\|u_{+}\|_{pr^{{}^{\prime}}}^{p}\leq\lambda\|V\|_{r}|\Omega_{+}|^{\frac{1}{r^{{}^{\prime}}}-\frac{p}{\alpha}}\|u_{+}\|_{\alpha}^{p}.$

On the other hand, by Sobolev embedding, $\|u_{+}\|_{\alpha}^{p}\leq
C\left[u_{+}\right]_{s,p}^{p}$, which gives

$1\leq\lambda
C\|V\|_{r}\left|\Omega_{+}\right|^{\frac{1}{r^{\prime}}-\frac{p}{\alpha}}$

and then the result holds for $|\Omega_{+}|$. The inequality for
$|\Omega_{-}|$ is obtained analogously. ∎

The following statement summarizes some basic facts about the first eigenvalue
$\lambda_{1}(p,V)$.

######  Proposition 3.3.

  1. (i)

$\lambda_{1}(p,V)$ is simple, and all eigenfunctions associated to an
eigenvalue $\lambda>\lambda_{1}(p,V)$ must change sign.

  2. (ii)

All eigenfunctions associated to $\lambda_{1}(p,V)$ have constant sign.

  3. (iii)

If $\Omega$ is a ball, then all eigenfunctions associated to
$\lambda_{1}(p,V)$ are radially symmetric and radially monotone.

  4. (iv)

$\lambda_{1}(p,V)$ is isolated.

###### Proof.

$(i)$. It follows by Picone’s identity.

$(ii)$. The eigenfunction $u_{1}$ can be chosen to be positive. Indeed, since

$||u_{1}(x)|-|u_{1}(y)||\leq|u_{1}(x)-u_{1}(y)|\ \text{for all}\
x,y\in\mathbb{R}^{N},$

we get $[|u_{1}|]_{s,p}^{p}\leq[u_{1}]_{s,p}^{p}\leq\lambda_{1}(p,V)$. This
implies that $|u_{1}|$ is an eigenfunction associated to $\lambda_{1}(p,V)$,
and by Proposition 3.2, we have $|u_{1}|>0$.

$(iii)$. Let $u$ be an eigenfunction associated to $\lambda_{1}$ in the ball
$\Omega$ and denote by $u^{*}$ the Schwarz symmetrization of $u$. By $(ii)$,
we may assume that $u>0$. By [7, Theorem 3] it is known that
$\|u^{*}\|_{p}=\|u\|_{p}$, $\|u^{*}\|_{V,p}=\|u\|_{V,p}$ and
$[u^{*}]_{s,p}\leq[u]_{s,p}$. Hence, we have from Proposition 3.1 that

$\lambda_{1}(p,V)\leq[u^{*}]_{s,p}^{p}+|u^{*}\|_{p}^{p}\leq[u]_{s,p}^{p}+|u\|_{p}^{p}=\lambda_{1}(p,V).$

Thus $u^{*}$ is an eigenfunction associated to $\lambda_{1}(p,V)$. By $(i)$,
$u^{*}$ and $u$ are proportional and by the inequality above we obtain
$u^{*}=u$.

$(iv)$. By the definition of the eigenvalue, $\lambda_{1}(p,V)$ is left-
isolated. To prove that $\lambda_{1}(p,V)$ is right-isolated, we argue by
contradiction. We assume that there exists a sequence of eigenvalues
$\\{\lambda_{k}\\}_{k\in\mathbb{N}}$ such that $\lambda_{k}>\lambda_{1}(p,V)$
and $\lambda_{k}\searrow\lambda_{1}(p,V)$ as $k\rightarrow\infty$. Let $u_{k}$
be an eigenfunction associated to $\lambda_{k}$, we can assume that
$\|u_{k}\|^{p}_{s,V}=1$. Then $\\{u_{k}\\}_{k\in\mathbb{N}}$ is bounded in
$W^{s,p}_{0}(\Omega)$ and up to a subsequence $u_{k}$ converges to $u$ weakly
in $W^{s,p}_{0}(\Omega)$ and strongly in $L^{p}(\Omega,V)$. Then
$\|u\|^{p}_{p,V}=1$ and

$\displaystyle[u]^{p}_{s,p}+\|u\|_{p}^{p}$
$\displaystyle\leq\liminf_{k\rightarrow\infty}([u_{k}]^{p}_{s,p}+\|u_{k}\|_{p}^{p})=\lim_{k\rightarrow\infty}\lambda_{k}\int_{\Omega}V(x)|u_{k}|^{p}\,dx$
$\displaystyle=\lambda_{1}(p,V)\int_{\Omega}V(x)|u|^{p}dx.$

Hence, $u$ is an eigenfunction associated to $\lambda_{1}(p,V)$. By $(ii)$, we
can assume that $u>0$. On the other hand, by the Egorov’s theorem, for any
$\varepsilon>0$ there exists $U_{\varepsilon}\subset\Omega$ such that
$|U_{\varepsilon}|<\varepsilon$ and $u_{k}\rightarrow u$ uniformly in
$\Omega\setminus U_{\varepsilon}$. This contradicts Lemma 3.1 since, for
$\alpha\in\left(pr^{\prime},p_{s}^{*}\right)$, we have

$0<\lim_{k\rightarrow\infty}(\lambda_{k}C\|V\|_{r})^{\frac{1}{\frac{p}{\alpha}-\frac{1}{r^{\prime}}}}\leq\left|\Omega_{\pm}\right|\leq\lim_{k\rightarrow\infty}|\\{x\in\mathbb{R}^{N}\colon\
u_{k}<0\\}|,$

getting a contradiction when $k\rightarrow\infty$. ∎

###  3.2. Sequence of eigenvalues

We now construct a suitable sequence of eigenvalues $(\lambda_{k})_{k}$ of
(5). The construction is based on the Alexander-Spanier Cohmology with
$\mathbb{Z}_{2}-$coefficients method (see Fadell and Rabinowitz [29]).

In order to neatly present our findings, we first need to give some basic
notations. Denote by $\mathcal{F}$ the class of symmetric subsets of $S$, and
denote by $i(A)$ the $\mathbb{Z}_{2}$-cohomological index of a symmetric set
$A\subset W^{s,p}_{0}(\Omega)\setminus\\{0\\}$. Let

$\mathcal{F}_{k}=\\{M\in\mathcal{F}:i(M)\geq k\\},$

We set for all $u\in W^{s,p}_{0}(\Omega)$

$I(u)=[u]_{s,p}^{p}+\|u\|_{p}^{p},\quad J(u)=\int_{\Omega}V(x)|u|^{p}dx,$
$S_{V}=\\{u\in W^{s,p}_{0}(\Omega):\ \int_{\Omega}V(x)|u|^{p}dx=1\\}.$

We denote by $\bar{I}$ the restriction of $I$ to $S_{V}$. For all $\lambda>0$,
$\lambda$ is a critical value of $\bar{I}$ if and only if it is an eigenvalue
of (5).

Now, we can construct the sequence $(\lambda_{k})_{k}$ of eigenvalues for
problem (5), setting

$\lambda_{k}=\inf_{A\in\mathcal{F}_{k}}\sup_{u\in A}I(u).$

Since $S_{V}=J^{-1}(1)$, then $u\in S_{V}$ is a critical point of $\bar{I}$ if
and only if there is some $\alpha\in\mathbb{R}$ such that
$I^{{}^{\prime}}(u)=\alpha J^{{}^{\prime}}(u)$.

######  Proposition 3.4.

The functional $\bar{I}$ satisfies the Palais-Smale condition at any level
$c\in\mathbb{R}$ and for all $k\in\mathbb{N}$, $\lambda_{k}$ is an eigenvalue
of problem (5). Moreover, $\lambda_{k}\rightarrow\infty$.

###### Proof.

Let $c\in\mathbb{R}$ and $(u_{n})_{n}\subset S_{V}$ be a $(PS)_{c}$ sequence
of $\bar{I}$. Hence there exists a suitable sequence
$(\alpha_{n})_{n}\subset\mathbb{R}$ such that

$\bar{I}(u_{n})\rightarrow
c\quad\text{and}\quad\bar{I}^{{}^{\prime}}(u_{n})-\alpha_{n}J^{{}^{\prime}}(u_{n})\rightarrow
0\ \text{in}\ W^{s,p}_{0}(\Omega).$

Clearly $(u_{n})_{n}$ is bounded in $W^{s,p}_{0}(\Omega)$. Therefore we can
pass to a subsequence $u_{n}\rightharpoonup u$ in $W^{s,p}_{0}(\Omega)$. Hence
$u_{n}\rightarrow u$ in $L^{p}(\Omega,V)$. In particular $u\in S_{V}$. Now,

$\bar{I}(u_{n})-\alpha_{n}=(\bar{I}^{{}^{\prime}}(u_{n}),u_{n})-\alpha_{n}(J^{{}^{\prime}}(u_{n}),u_{n})\rightarrow
0.$

Hence

$\alpha_{n}=\bar{I}(u_{n})+o(1)\rightarrow c.$

Notice that, for all $n\in\mathbb{N}$, we have

$\displaystyle|(\bar{I}^{{}^{\prime}}(u_{n}),u_{n}-u)|$
$\displaystyle=(A_{p}u_{n},u_{n}-u)+\int_{\Omega}|u_{n}|^{p-2}u_{n}(u_{n}-u)dx$
$\displaystyle=\alpha_{n}\left(\int_{\Omega}V(x)|u_{n}|^{p-2}u_{n}(u_{n}-u)\right)+o(1)$
$\displaystyle\leq\alpha_{n}C\|\omega_{n}\|_{V,p^{{}^{\prime}}}\|u_{n}-u\|_{p},$

and the latter vanishes as $n\rightarrow\infty$ because
$\|\omega_{n}\|_{V,p^{{}^{\prime}}}\rightarrow\|\omega\|_{V,p^{{}^{\prime}}}$
where $\omega_{n}=|u_{n}|^{p-2}u_{n}$ and and similarly $\omega=|u|^{p-2}u$.
Hence, by the $(S)$-property of $A_{p}$, we get $u_{n}\rightarrow u$ in
$W^{s,p}_{0}(\Omega)$.In other words, $\bar{I}$ verifies the $(PS)_{c}$
condition for all $c\in\mathbb{R}^{+}$.

An application of [42, Theorem 4.6] to the even functional $I$ shows that
$\lambda_{k}\rightarrow\infty$ as $k\rightarrow\infty$. ∎

###  3.3. The second eigenvalue

We start by defining

$\lambda_{2}(\Omega)=\inf_{f\in
C_{1}(\Omega)}\max_{u\in\operatorname{lm}(f)}I(u)$

where the set $\mathcal{C}_{1}(\Omega)$ is given by

$\mathcal{C}_{1}(\Omega)=\left\\{f:\mathbb{S}^{1}\rightarrow\mathcal{S}_{V}:f\text{
odd and continuous }\right\\}$

######  Theorem 3.1.

The quantity $\lambda_{2}(\Omega)$ is an eigenvalue. Moreover we have
$\lambda_{1}(p,V)<\lambda_{2}(\Omega)$ and every eigenfunction $u\in S_{V}$
associated to $\lambda_{2}(\Omega)$ has to change sign.

###### Proof.

Let us prove that $\lambda_{2}(\Omega)$ is an eigenvalue. In order to show
that $\lambda_{2}(\Omega)$ is a critical point of the functional $I$ defined
on the manifold $S_{V}$, it is sufficient to check that $I$ verifies the
Palais-Smale condition. The claim will then follow by Proposition 3.4.

Now, we prove that $\lambda_{2}(\Omega)>\lambda_{1}(p,V).$ Let us argue by
contradiction and suppose that

$\lambda_{2}(\Omega)=\inf_{f\in\mathcal{C}_{1}(\Omega)}\max_{u\in\operatorname{lm}(f)}I(u)=\lambda_{1}(p,V)$

so that, for all $n\in\mathbb{N}$ there exists an odd continuous mapping
$f_{n}:\mathbb{S}^{1}\rightarrow S_{V}$ such that

(8) $\max_{u\in f_{n}\left(S^{1}\right)}I(u)\leq\lambda_{1}(p,V)+\frac{1}{n}.$

Let us denote by $u_{1}$ the global minimizer of $I$ belonging to
$\mathcal{S}_{V}$. From Proposition 3.3 $(ii)$, $u_{1}$ must be nonnegative
(or nonpositive). Let $0<\varepsilon\ll 1$ and consider the two neighborhoods

$\mathcal{B}_{\varepsilon}^{+}=\left\\{u\in\mathcal{S}_{V}:\left\|u-u_{1}\right\|_{L^{p}(\Omega)}<\varepsilon\right\\}$
$\mathcal{B}_{\varepsilon}^{-}=\left\\{u\in\mathcal{S}_{V}:\left\|u-\left(-u_{1}\right)\right\|_{L^{p}(\Omega)}<\varepsilon\right\\}$

which are disjoint, by construction. Since the mapping $f_{n}$ is odd and
continuous, for every $n\in\mathbb{N}$ the image
$f_{n}\left(\mathbb{S}^{1}\right)$ is symmetric and connected, then it can not
be contained in
$\mathcal{B}_{\varepsilon}^{+}\cup\mathcal{B}_{\varepsilon}^{-}$, the latter
being symmetric and disconnected. So we can pick an element

(9) $u_{n}\in
f_{n}\left(S^{1}\right)\backslash\left(\mathcal{B}_{\varepsilon}^{+}\cup\mathcal{B}_{\varepsilon}^{-}\right)$

This yields a sequence
$\left\\{u_{n}\right\\}_{n\in\mathbb{N}}\subset\mathcal{S}_{V}$, which is
bounded in $W_{0}^{s,p}(\Omega)$ by (8). Hence, there exists a function $v\in
S_{V}$ such that $\left\\{u_{n}\right\\}_{n\in\mathbb{N}}$ converges to $v$
weakly in $W_{0}^{s,p}(\Omega)$ and strongly in $L^{p}(\Omega)$, possibly by
passing to a subsequence. By the weak convergence of $I$ it follows that

$I(v)\leq\liminf_{n\rightarrow\infty}I(u_{n})=\lambda_{1}(\Omega)$

This in turn shows that $v\in\mathcal{S}_{V}$ is a global minimizer, so that
either $v=u_{1}$ or $v=-u_{1}$. On the other hand, by strong $L^{\text{p }}$
convergence we also have

$v\in\mathcal{S}_{V}\backslash\left(B_{\varepsilon}^{+}\cup
B_{\varepsilon}^{-}\right).$

This gives a contradiction and thus $\lambda_{2}(\Omega)>\lambda_{1}(p,V)$.

Finally, it follows, directly from the previous assertion and Proposition 3.3
$(i)$, that $\lambda_{2}(\Omega)$ admits only sign-changing eigenfunctions.

∎

######  Proposition 3.5.

For every eigenvalue $\lambda>\lambda_{1}(p,V)$ we have that
$\lambda\geq\lambda_{2}(\Omega)$. In particular $\lambda_{1}(p,V)$ is
isolated.

###### Proof.

Let $\lambda>\lambda_{1}(p,V)$ be an eigenvalue and $u$ be a corresponding
eigenfunction. By Proposition 3.3 $(i)$, $u$ must change sign. Hence, we
assume that $u=u_{+}+u_{-}$ where $u_{+}=\max\\{u,0\\}\neq 0$,
$u_{-}=\min\\{u,0\\}\neq 0$. By testing the equation solved by $u$ against
$u_{+}$ and $u_{-}$, we get

$\displaystyle\lambda\int_{\Omega}V(x)|u_{+}|^{p}dx=$ $\displaystyle
K(N,s,p)\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}(u_{+}(x)-u_{+}(y))dxdy+\int_{\Omega}|u_{+}|^{p}dx,$

and

$\displaystyle\lambda\int_{\Omega}V(x)|u_{-}|^{p}dx=$ $\displaystyle
K(N,s,p)\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}(u_{-}(x)-u_{-}(y))dxdy+\int_{\Omega}|u_{-}|^{p}dx,$

which can be rewritten as

$\lambda\int_{\Omega}V(x)|u_{+}|^{p}dx=K(N,s,p)\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|U+W|^{p-2}(U+W)}{|x-y|^{N+ps}}Udxdy+\int_{\Omega}|u_{+}|^{p}dx,$

and

$\lambda\int_{\Omega}V(x)|u_{-}|^{p}dx=K(N,s,p)\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|U+W|^{p-2}(U+W)}{|x-y|^{N+ps}}Wdxdy+\int_{\Omega}|u_{-}|^{p}dx,$

where $U(x,y)=u_{+}(x)-u_{+}(y)$ and $W(x,y)=u_{-}(x)-u_{-}(y)$. Let
$(\theta_{1},\theta_{2})\in\mathbb{S}^{1}$, by multiplying the previous two
identities by $|\theta_{1}|^{p}$ and $|\theta_{2}|^{p}$ and adding them, we
can arrive at

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{K(N,s,p)\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{[|\theta_{1}|^{p}|U+W|^{p-2}(U+W)U+|\theta_{2}|^{p}|U+W|^{p-2}(U+W)W}{|x-y|^{N+ps}}dxdy}{|\theta_{1}|^{p}\int_{\Omega}V(x)|u_{+}|^{p}dx+|\theta_{2}|^{p}\int_{\Omega}V(x)|u_{-}|^{p}dx}$
$\displaystyle+\frac{|\theta_{1}|^{p}\int_{\Omega}|u_{+}|^{p}dx+|\theta_{2}|^{p}\int_{\Omega}|u_{-}|^{p}dx}{|\theta_{1}|^{p}\int_{\Omega}V(x)|u_{+}|^{p}dx+|\theta_{2}|^{p}\int_{\Omega}V(x)|u_{-}|^{p}dx}.$

Now we define the following element of $\mathcal{C}_{1}(\Omega)$

$f(\theta)=\frac{\theta_{1}u_{+}+\theta_{2}u_{-}}{(|\theta_{1}|^{p}\int_{\Omega}V(x)|u_{+}|^{p}dx+|\theta_{2}|^{p}\int_{\Omega}V(x)|u_{-}|^{p}dx)^{1/p}},\theta=(\theta_{1},\theta_{2})\in\mathbb{S}^{1}.$

Then

$\displaystyle I(f(\theta))$ $\displaystyle=$
$\displaystyle\frac{K(N,s,p)\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|\theta_{1}u_{+}(x)+\theta_{2}u_{-}(x)-(\theta_{1}u_{+}(y)+\theta_{2}u_{-}(y))|^{p}}{|x-y|^{N+ps}}dxdy}{|\theta_{1}|^{p}\int_{\Omega}V(x)|u_{+}|^{p}dx+|\theta_{2}|^{p}\int_{\Omega}V(x)|u_{-}|^{p}dx}$
$\displaystyle+\frac{\int_{\Omega}|\theta_{1}u_{+}+\theta_{2}u_{-}|^{p}dx}{|\theta_{1}|^{p}\int_{\Omega}V(x)|u_{+}|^{p}dx+|\theta_{2}|^{p}\int_{\Omega}V(x)|u_{-}|^{p}dx}.$
$\displaystyle=$
$\displaystyle\frac{K(N,s,p)\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|\theta_{1}U+\theta_{2}W|^{p}}{|x-y|^{N+ps}}dxdy}{|\theta_{1}|^{p}\int_{\Omega}V(x)|u_{+}|^{p}dx+|\theta_{2}|^{p}\int_{\Omega}V(x)|u_{-}|^{p}dx}$
$\displaystyle+\frac{|\theta_{1}|^{p}\int_{\Omega}|u_{+}|^{p}dx+|\theta_{2}|^{p}\int_{\Omega}|u_{-}|^{p}dx}{|\theta_{1}|^{p}\int_{\Omega}V(x)|u_{+}|^{p}dx+|\theta_{2}|^{p}\int_{\Omega}V(x)|u_{-}|^{p}dx}.$

By (4.7) in the proof of [19, Proposition 4.2], we have

$\displaystyle|\theta_{1}|^{p}|U+W|^{p-2}(U+W)U+|\theta_{2}|^{p}|U+W|^{p-2}(U+W)W$
$\displaystyle=$
$\displaystyle|\theta_{1}U+\theta_{1}W|^{p-2}(\theta_{1}U+\theta_{1}W)\theta_{1}U+|\theta_{2}U+\theta_{2}W|^{p-2}(\theta_{2}U+\theta_{2}W)\theta_{2}W$
$\displaystyle\geq$ $\displaystyle|\theta_{1}U+\theta_{2}W|^{p}.$

Hence,

$I(f(\theta))\leq\lambda,\forall\theta\in\mathbb{S}^{1}.$

By appealing to the definition of $\lambda_{2}(\Omega)$ we get the desired
conclusion. Noticing that $\lambda_{1}(p,V)<\lambda_{2}(\Omega)$, we know that
$\lambda_{1}(p,V)$ is isolated. ∎

######  Corollary 3.1.

$\lambda_{2}(\Omega)=\lambda_{2}$ where $\lambda_{2}$ is defined by
$\mathbb{Z}_{2}$-cohomological index.

###### Proof.

Obviously, $\lambda_{2}(\Omega)\leq\lambda_{2}$ since
$\lambda_{2}>\lambda_{1}(p,V)$ is an eigenvalue. Next we prove that
$\lambda_{2}(\Omega)\geq\lambda_{2}$

For any $f\in\mathcal{C}_{1}(\Omega)$, we define
$M=\\{f(\theta):\theta\in\mathbb{S}^{1}\\}$, which is a symmetric set. It is
not difficult to verify that $i(M)\geq 2$. Thus

$\sup_{\theta\in\mathbb{S}^{1}}I(f(\theta))=\sup_{u\in
M}I(u)\geq\inf_{A\in\mathcal{F}_{2}}\sup_{u\in A}I(u)=\lambda_{2},\forall
f\in\mathcal{C}_{1}(\Omega).$

Hence,

$\lambda_{2}(\Omega)=\inf_{f\in\mathcal{C}_{1}(\Omega)}\sup_{\theta\in\mathbb{S}^{1}}I(f(\theta))\geq\lambda_{2},$

and we complete the proof. ∎

##  4\. The Case $\mu>0$ versus a discontinuity spectrum

######  Proposition 4.1.

For $\mu>0$, we set

(10)
$\lambda\left(p,q,\mu,V\right):=\inf\left\\{\frac{\mathcal{I}(u)}{\int_{\Omega}V(x)|u|^{p}dx}:u\in\mathcal{W}^{s},\int_{\Omega}V(x)|u|^{p}dx>0\right\\},$

where

$\mathcal{I}(u)=\mathcal{J}_{s}(u)+\frac{1}{p}\|u\|_{p}^{p}+\frac{\mu}{q}\|u\|_{q}^{q}.$

Then,

$\lambda\left(p,q,\mu,V\right)=\lambda_{1}\left(p,V\right)$

holds for every $\mu>0.$ In addition, for every $\mu>0$, the infimum in (10)
is not attained.

###### Proof.

Fix $\mu>0$. From the definitions of $\lambda_{1}\left(p,V\right)$ and
$\lambda\left(p,q,\mu,V\right)$, we can see that
$\lambda\left(p,q,\mu,V\right)\geq\lambda_{1}\left(p,V\right)$. Let $u_{1}$ be
the positive eigenfunction corresponding to $\lambda_{1}\left(p,V\right)$ such
that $\|u_{1}\|_{p,V}=1$. Thus, $u_{1}$ satisfies
$\|u_{1}\|_{s,p}^{p}=\lambda_{1}\left(p,V\right)$.

By using the homogeneity of the norms and letting $t\rightarrow\infty$ in the
following inequality

$\lambda\left(p,q,\mu,V\right)\leq\frac{\|tu_{1}\|_{s,p}^{p}+\frac{\mu}{q}\|tu_{1}\|_{s,q}^{q}}{\|tu_{1}\|_{p,V}^{p}}=\lambda_{1}(p,V)+\frac{\mu}{q}\frac{\|u_{1}\|_{s,q}^{q}}{t^{p-q}}.$

We obtain $\lambda\left(p,q,\mu,V\right)\leq\lambda_{1}\left(p,V\right)$ since
$p>q$.

We now prove that $\lambda\left(p,q,\mu,V\right)$ is not attained for any
$\mu>0$. Assume that there exist $\mu>0$ and a function $u\in\mathcal{W}^{s}$
such that $\|u\|_{p,V}>0$ and

$\lambda\left(p,q,\mu,V\right)=\frac{\|u\|_{s,p}^{p}+\frac{\mu}{q}\|u\|_{s,q}^{q}}{\|u\|_{p,V}^{p}}$

Then, since $u\neq 0$, the definition of $\lambda_{1}\left(p,V\right)$ (note
that $\mathcal{W}^{s}\subset W_{0}^{s,q}(\Omega)$ and the first assertion of
the proposition, that

$\displaystyle\lambda\left(p,q,\mu,V\right)=\frac{\|u\|_{s,p}^{p}+\frac{\mu}{q}\|u\|_{s,q}^{q}}{\|u\|_{p,V}^{p}}\geq\lambda_{1}(p,V)+\frac{\mu}{q}\frac{\|u\|_{s,q}^{q}}{\|u\|_{p,V}^{p}}$
$\displaystyle>\lambda_{1}\left(p,V\right)=\lambda\left(p,q,\mu,V\right).$

This is a contradiction. ∎

###### Proof of Theorem 1.1.

$(i)$ Fix $\mu>0.$ Let $u$ be a non-trivial solution of (2). By taking $u$ as
test function in (2) and since $\mu>0$ we have

$0<[u]_{s,p}^{p}<[u]_{s,p}^{p}+\mu[u]_{s,q}^{q}=\lambda\int_{\Omega}V|u|^{p}dx.$

This yields that one of the following alternatives $(a)$ or $(b)$ occurs:

  1. (a)

$\lambda<0$ and $\int_{\Omega}V|u|^{p}dx<0$ or

  2. (b)

$\lambda>0$ and $\int_{\Omega}V|u|^{p}dx>0$.

It is sufficient to only consider the case $(b)$ because we may consider the
pair of $-\lambda$ and $-V$ instead of $\lambda$ and $V$ provided $V$ changes
sign. Thus, we may assume that $\int_{\Omega}V|u|^{p}dx>0$ and $\lambda>0$,
whence we have

$\lambda_{1}\left(p,V\right)\leq\frac{[u]_{s,p}^{p}}{\|u\|_{V,p}^{p}}<\frac{[u]_{s,p}^{p}+\mu[u]_{s,q}^{q}}{\|u\|_{p,V}^{p}}=\lambda$

by the definition of $\lambda_{1}\left(p,V\right)$. The first assertion
follows.

$(ii)$ Arguing by contradiction, we assume that (2) has a sign-changing
solution $u\in\mathcal{W}^{s}$ for some $\mu>0$ and $\lambda$ satisfying
$\lambda_{1}\left(p,V\right)<\lambda\leq\lambda_{2}\left(p,V\right)$. Then,
taking $\pm u_{\pm}$as test functions, we obtain

$\displaystyle
0<[u_{+}]_{s,p}^{p}<[u_{+}]_{s,p}^{p}+\mu[u_{+}]_{s,q}^{q}\leq\lambda\int_{\Omega}Vu_{+}^{p}dx\quad\text{
and }$ $\displaystyle
0<[u_{-}]_{s,p}^{p}<[u_{-}]_{s,p}^{p}+\mu[u_{-}]_{s,q}^{q}\leq\lambda\int_{\Omega}Vu_{-}^{p}dx$

where $u_{\pm}:=\max\\{\pm u,0\\}.$ Note that $u_{\pm}\in\mathcal{W}^{s}$ and
$J\left(u_{\pm}\right)=\int_{\Omega}Vu_{\pm}^{p}dx>0$ because $\lambda>0$.
Combining these inequalities and the argument in [41, Proposition 11], we can
construct a continuous path $\gamma_{0}\in\mathcal{C}_{1}(\Omega)$ such that

$\max_{t\in\mathbb{S}^{1}}\mathcal{I}\left(\gamma_{0}(t)\right)<\lambda.$

This leads to $\lambda_{2}(p,V)<\lambda$, which contradicts
$\lambda\leq\lambda_{2}(p,V)$.

∎

The proof of assertion $(iii)$ of Theorem 1.1 will be divided into several
lemmas. Let first consider the Nehari manifold

$\displaystyle\mathcal{N}_{\lambda}$
$\displaystyle:=\left\\{u\in\mathcal{W}^{s}\backslash\\{0\\}:\left\langle\mathcal{J}^{\prime}(u),u\right\rangle=0\right\\}$
$\displaystyle=\left\\{u\in\mathcal{W}^{s}\backslash\\{0\\}:[u]_{s,p}^{p}+\mu[u]_{s,q}^{q}+\int_{\Omega}\left(|u|^{p}+\mu|u|^{q}\right)dx=\lambda\int_{\Omega}V|u|^{p}dx\right\\}$

Note that for each $u\in\mathcal{N}_{\lambda}$ the functional $\mathcal{J}$
has the following expression

(11)
$\mathcal{J}(u)=\mu\left(\frac{1}{q}-\frac{1}{p}\right)\left([u]_{s,q}^{q}+\|u\|_{q}^{q}\right)\geq
0.$

######  Lemma 4.1.

$\mathcal{N}_{\lambda}\neq\emptyset$.

###### Proof.

Since $\lambda>\lambda_{1}(p,V)$, then there exists
$w\in\mathcal{W}^{s}\backslash\\{0\\}$ for which

$\lambda\int_{\Omega}V|\omega|^{p}dx>[\omega]_{s,p}^{p}+\|\omega\|_{p}^{p}.$

Consider the function $\xi\colon(0,+\infty)\rightarrow\mathbb{R}$ defined by

$\displaystyle\xi(t)$
$\displaystyle=\left\langle\mathcal{J}^{\prime}(t\omega),t\omega\right\rangle$
$\displaystyle=t^{p}[\omega]_{s,p}^{p}+\mu
t^{q}[\omega]_{s,q}^{q}+t^{p}\int_{\Omega}|\omega|^{p}dx+\mu
t^{q}\int_{\Omega}|\omega|^{q}dx-\lambda t^{p}\int_{\Omega}V|\omega|^{p}dx$
(12)
$\displaystyle=t^{q}\mu\left([\omega]_{s,q}^{q}+\int_{\Omega}|\omega|^{q}dx\right)-t^{p}\left(\int_{\Omega}V|\omega|^{p}dx-[\omega]_{s,p}^{q}-\int_{\Omega}|\omega|^{p}dx\right)$

Since $p>q$, we see from (4) that

$\xi(t)\rightarrow-\infty\quad\text{as}\ t\rightarrow+\infty.$

On the other hand, for small $t\in(0,1)$ we have

$\xi(t)>0.$

Therefore, we can find $t_{0}>0$ such that $\xi(t_{0})=0$ then
$t_{0}\omega\in\mathcal{N}_{\lambda}$ and so
$\mathcal{N}_{\lambda}\neq\emptyset$. ∎

We define

(13) $m_{\lambda}=\inf\\{\mathcal{J}(u)\colon u\in\mathcal{N}_{\lambda}\\}.$

By (11) we deduce that $m_{\lambda}\geq 0$ for all
$u\in\mathcal{N}_{\lambda}$.

######  Lemma 4.2.

  1. (i)

Every minimizing sequence of (13) is bounded in $\mathcal{W}^{s}$.

  2. (ii)

$m_{\lambda}>0$.

###### Proof.

$(i)$ We argue by contradiction. So, suppose that
$(u_{n})\subset\mathcal{W}^{s}$ is a minimizing sequence of (13) such that

$[u_{n}]_{s}\rightarrow+\infty.$

Without loss of generality we can consider

$[u_{n}]_{s,p}\rightarrow+\infty.$

We have

(14)
$[u_{n}]_{s,p}^{p}+\mu[u_{n}]_{s,q}^{q}+\int_{\Omega}\left(|u_{n}|^{p}+\mu|u_{n}|^{q}\right)dx=\lambda\int_{\Omega}V|u_{n}|^{p}dx,$

and

(15)
$[u_{n}]_{s,p}^{p}\leq\lambda\|u_{n}\|_{V,p}^{p}\quad\Rightarrow\quad\|u_{n}\|_{V,p}\rightarrow+\infty.$

We set $\omega_{n}=\frac{u_{n}}{\|u_{n}\|_{V,p}}$ for all $n\in\mathbb{N}$,
hence $\|\omega_{n}\|_{V,p}=1$. Also, from (15) we have

$[\omega_{n}]_{s,p}^{p}\leq\lambda\quad\text{for all}\ n\in\mathbb{N},$

so $(\omega_{n})\subset\mathcal{W}^{s}$ is bounded and we may assume that

$\omega_{n}\rightharpoonup\omega\ \text{in}\
\mathcal{W}^{s}\quad\text{and}\quad\omega_{n}\rightarrow\omega\ \text{in}\
L^{p}(V,\Omega).$

Recall that $(u_{n})\subset\mathcal{W}^{s}$ is a minimizing sequence for (13).
So, we have

(16)
$\left([u_{n}]_{s,q}^{q}+\|u_{n}\|_{q}^{q}\right)\rightarrow\frac{1}{\mu}\left(\frac{1}{q}-\frac{1}{p}\right)^{-1}m_{\lambda}\
\text{as}\ n\rightarrow\infty.$

We multiply (14) with $1/\|u_{n}\|_{V,p}$, we obtain

(17)
$\mu\left([\omega_{n}]_{s,q}^{q}+\|\omega_{n}\|_{q}^{q}\right)=\lambda-[\omega_{n}]_{s,p}^{p}-\|\omega_{n}\|_{p}^{p}.$

Using (16) in (17), we can infer that $\omega=0$. On the other hand, since
$\|\omega_{n}\|_{V,p}=1$ for each $n$ we get that $\|\omega\|_{V,p}=1$ which
is a contradiction with $\omega=0$. Therefore we can conclude that every
minimizing sequence of (13) is bounded in $\mathcal{W}^{s}$.

$(ii)$ We already observed that $m_{\lambda}\geq 0$. Suppose by contradiction
that $m_{\lambda}=0$. Let $(u_{n})\subset\mathcal{W}^{s}$ is a minimizing
sequence of (13). By similar arguments as the previous proof we show in one
hand that $\omega_{n}\rightarrow 0$ in $L^{p}(V,\Omega)$ and on other hand
$\omega_{n}=\frac{u_{n}}{\|u_{n}\|^{p}_{V,p}}\rightarrow 1$ which gives a
contradiction and we have the result.

∎

######  Lemma 4.3.

There exists $u\in\mathcal{N}_{\lambda}$ such that
$\mathcal{J}(u)=m_{\lambda}$.

###### Proof.

Let $\left\\{u_{n}\right\\}_{n\geqslant 1}\subseteq\mathcal{N}_{\lambda}$ be
such that $\mathcal{J}\left(u_{n}\right)\rightarrow m_{\lambda}$. According to
Lemma 4.2, $\left\\{u_{n}\right\\}_{n\geqslant 1}\subseteq\mathcal{W}^{s}$ is
bounded. So, we may assume that

(18) $u_{n}\rightharpoonup\hat{u}_{\lambda}\quad\text{ in
}\mathcal{W}^{s}\quad\text{ and }\quad
u_{n}\rightarrow\hat{u}_{\lambda}\quad\text{ in }L^{p}(V,\Omega).$

Since $u_{n}\in\mathcal{N}_{\lambda}$ for all $n\in\mathbb{N}$, we have

(19)
$[u_{n}]_{s,p}^{p}+\mu[u_{n}]_{s,q}^{q}+\int_{\Omega}\left(|u_{n}|^{p}+\mu|u_{n}|^{q}\right)dx=\lambda\int_{\Omega}V|u_{n}|^{p}dx,$

Passing to the limit as $n\rightarrow\infty$ and using (18) and the weak lower
semicontinuity of the norm functional in a Banach space, we obtain

(20)
$\mu\left([\hat{u}_{\lambda}]_{s,q}^{q}+\|\hat{u}_{\lambda}\|_{q}^{q}\right)dx\leq\lambda\|\hat{u}_{\lambda}\|_{V,p}-[\hat{u}_{\lambda}]_{s,p}^{p}+\|\hat{u}_{\lambda}\|_{p}^{p}.$

Note that
$\lambda\|\hat{u}_{\lambda}\|_{V,p}-[\hat{u}_{\lambda}]_{s,p}^{p}+\|\hat{u}_{\lambda}\|_{p}^{p}\neq
0$, or otherwise from (17), we have

$[u_{n}]_{s,q}^{q}+\|u_{n}\|_{q}^{q}\rightarrow 0,$

so $u_{n}\rightarrow 0$ in $\mathcal{W}^{s}.$ Recall that

$\mathcal{J}(u_{n})=\mu\left(\frac{1}{q}-\frac{1}{p}\right)\left([u_{n}]_{s,q}^{q}+\|u_{n}\|_{q}^{q}\right)\quad\text{
for all }n\in\mathbb{N}.$

So, it follows that

$\mathcal{J}(u_{n})\rightarrow 0\quad\text{ as
}n\rightarrow\infty\quad\text{and}\quad m_{\lambda}=0,$

which contradicts Lemma 4.2. Therefore

$\lambda\|\hat{u}_{\lambda}\|_{V,p}-[\hat{u}_{\lambda}]_{s,p}^{p}+\|\hat{u}_{\lambda}\|_{p}^{p}\neq
0\quad\text{and}\quad\hat{u}_{\lambda}\neq 0.$

Also, exploiting the sequential weak lower semicontinuity of $\mathcal{J}$, we
have

$\mathcal{J}\left(\hat{u}_{\lambda}\right)\leqslant\lim_{n\rightarrow\infty}\mathcal{J}\left(u_{n}\right)=m_{\lambda}$

If we show that $\hat{u}_{\lambda}\in\mathcal{N}_{\lambda}$, then
$\mathcal{J}\left(\hat{u}_{\lambda}\right)=m_{\lambda}$, and this will
conclude the proof. To this end, let

$\hat{\xi}_{\lambda}(t)=\left\langle\mathcal{J}^{\prime}\left(t\hat{u}_{\lambda},t\hat{u}_{\lambda}\right)\right\rangle\quad\text{
for all }t\in[0,1].$

Evidently, $\hat{\xi}_{\lambda}(\cdot)$ is a continuous function. Arguing by
contradiction, suppose that $\hat{u}_{\lambda}\notin\mathcal{N}_{\lambda}$.
Then since $u_{n}\in\mathcal{N}_{\lambda}$ for all $n\in\mathbb{N}$, we can
infer from (20) that

(21) $\hat{\xi}_{\lambda}(1)<0.$

On the other hand, note that since $\lambda>\lambda_{1}(p,V)$ and from (4), we
have

$\hat{\xi}_{\lambda}(t)\geqslant c_{1}t^{q}-c_{2}t^{p}\quad\text{ for some
}c_{1},c_{2}>0.$

So, since $q<p$, we have

(22) $\hat{\xi}_{\lambda}(t)>0\quad\text{ for all}\quad
t\in(0,\epsilon)\quad\text{ with small}\ \epsilon\in(0,1).$

By (21) and (22), we see that there exists $t^{*}\in(0,1)$ such that

$\hat{\xi}_{\lambda}\left(t^{*}\hat{u}_{\lambda}\right)=0\quad\text{and}\quad
t^{*}\hat{u}_{\lambda}\in\mathcal{N}_{\lambda}.$

Then using (11), we have

$\displaystyle
m_{\lambda}\leqslant\varphi_{\lambda}\left(t^{*}\hat{u}_{\lambda}\right)$
$\displaystyle=\alpha\left(\frac{1}{q}-\frac{1}{p}\right)\left(t^{*}\right)^{q}[\hat{u}_{\lambda}]_{s,q}^{q}+\|\hat{u}_{\lambda}\|_{q}^{q}$
$\displaystyle<\alpha\left(\frac{1}{q}-\frac{1}{p}\right)[\hat{u}_{\lambda}]_{s,q}^{q}+\|\hat{u}_{\lambda}\|_{q}^{q}\quad\left(\text{since
}t^{*}\in(0,1)\right)$
$\displaystyle\leqslant\alpha\left(\frac{1}{q}-\frac{1}{p}\right)\liminf_{n\rightarrow\infty}\left([u_{n}]_{s,q}^{q}+\|u_{n}\|_{q}^{q}\right)$
$\displaystyle=m_{\lambda}$

a contradiction. Therefore $\hat{u}_{\lambda}\in\mathcal{N}_{\lambda}$, and
this finishes the proof. ∎

##  5\. Proofs of Stability results

First, we give a convergence principle of solutions of a more general type
equation with this form

(23) $\left\\{\begin{aligned}
(-\Delta)_{p}^{s}\,u+\mu(-\Delta)_{q}^{s}\,u+|u|^{p-2}u+\mu|u|^{q-2}u=f(x,u)\quad&\text{in
}\Omega\\\
u=0\quad&\text{in}\quad\mathbb{R}^{N}\backslash\Omega,\end{aligned}\right.$

Here, we need to impose some conditions on $f$ in order to guarantee that weak
solutions are well defined:

  1. ($f_{1}$)

$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function,
i.e. $f(\cdot,z)$ is measurable for any $z\in\mathbb{R}$ and $f(x,\cdot)$ is
continuous a.e. $x\in\Omega$.

  2. ($f_{2}$)

There exist a constant $C>0$ such that $|f(x,z)|\leq C(1+|z|)^{r-1}$ for some
$r\in\left[1,p_{s}^{*}\right)$.

We will prove that any accumulation point of the sequence of solutions is in
fact a solution to the local limit problem

(24) $\left\\{\begin{aligned}
(-\Delta)_{p}\,u+\mu(-\Delta)_{q}\,u+|u|^{p-2}u+\mu|u|^{q-2}u=f(x,u)\quad&\text{in
}\Omega\\\ u=0\quad&\text{in}\quad\partial\Omega.\end{aligned}\right.$

######  Theorem 5.1.

Let $0<s_{k}\rightarrow 1$ and let $u_{k}\in\mathcal{W}^{s_{k}}$ be a sequence
of solutions of (23) such that
$\sup_{k\in\mathbb{N}}\left[u_{k}\right]_{s_{k}}<\infty.$ Then, any
accumulation point $u$ of the sequence
$\left\\{u_{k}\right\\}_{k\in\mathbb{N}}$ in $L^{p}(\Omega)$ verifies that
$u\in W_{0}^{1,p}(\Omega)$ and it is a weak solution of (24).

###### Proof.

Assume that $u_{k}\rightarrow u$ in $L^{p}(\Omega)$. Then, since
$\left\\{u_{k}\right\\}_{k\in\mathbb{N}}$ is uniformly bounded in
$\mathcal{W}^{s_{k}}$, by [17] we obtain that $u\in W_{0}^{1,p}(\Omega)$.
Passing to a subsequence, if necessary, we can also assume that
$u_{k}\rightarrow u$ a.e. in $\Omega$.

On the other hand, if we define

$\eta_{1,k}:=\left(-\Delta_{p}\right)^{s_{k}}u_{k}\in\left(\mathcal{W}^{s_{k}}\right)^{\prime}\subset
W^{-1,p^{\prime}}(\Omega)$

and

$\eta_{2,k}:=\left(-\Delta_{q}\right)^{s_{k}}u_{k}\in
W^{-s_{k},q^{\prime}}(\Omega)\subset W^{-1,p^{\prime}}(\Omega)\subset
W^{-1,q^{\prime}}(\Omega),$

then $\left\\{\eta_{1,k}\right\\}_{k\in\mathbb{N}}$ and
$\left\\{\eta_{2,k}\right\\}_{k\in\mathbb{N}}$ are bounded in
$W^{-1,p^{\prime}}(\Omega)$ and $W^{-1,q^{\prime}}(\Omega)$ respectively, and
hence, up to a subsequence, there exists $\eta\in W^{-1,p^{\prime}}(\Omega)$
such that

$\eta_{k}:=\eta_{1,k}+\eta_{2,k}\rightarrow\eta\quad\text{ weakly in}\quad
W^{-1,p^{\prime}}(\Omega).$

Since $u_{k}$ solves (23), for any $v\in C_{c}^{\infty}(\Omega)$

$0=\left\langle\left(-\Delta_{p}\right)^{s_{k}}u_{k}+\mu\left(-\Delta_{q}\right)^{s_{k}}u_{k},v\right\rangle-\int_{\Omega}f\left(x,u_{k}\right)vdx+\int_{\Omega}(|u_{k}|^{p-2}u_{k}+\mu|u_{k}|^{q-2}u_{k})vdx.$

Using the convergence, taking the limit $k\rightarrow\infty$ we get

$0=\langle\eta,v\rangle-\int_{\Omega}f(x,u)vdx+\int_{\Omega}(|u|^{p-2}u+\mu|u|^{q-2}u)vdx.$

We want to identify $\eta$, more precisely, we will prove that

(25)
$\langle\eta,v\rangle=\left\langle-\Delta_{p}u-\mu\Delta_{p}u,v\right\rangle.$

For that purpose we use the monotonicity of the operator and the fact that
$u_{k}$ is solution of (23), Indeed,

$\displaystyle 0$
$\displaystyle\leq\left\langle\left(-\Delta_{p}\right)^{s_{k}}u_{k}+\mu\left(-\Delta_{q}\right)^{s_{k}}u_{k},u_{k}-v\right\rangle-\left\langle\left(-\Delta_{p}\right)^{s_{k}}v+\mu\left(-\Delta_{q}\right)^{s_{k}}v,u_{k}-v\right\rangle$
$\displaystyle=\int_{\Omega}f\left(x,u_{k}\right)\left(u_{k}-v\right)dx-\left\langle\left(-\Delta_{p}\right)^{s_{k}}v+\mu\left(-\Delta_{q}\right)^{s_{k}}v,u_{k}-v\right\rangle.$

Hence taking the limit $k\rightarrow\infty$ and using Lemma 2.2 one finds that

$\displaystyle 0$
$\displaystyle\leq\int_{\Omega}f(x,u)(u-v)-\left\langle-\Delta_{p}v-\mu\Delta_{q}v,u-v\right\rangle$
$\displaystyle=\langle\eta,u-v\rangle-\left\langle-\Delta_{p}v-\Delta_{q}v,u-v\right\rangle.$

Consequently, if we take $v=u-tw,w\in W_{0}^{1,p}(\Omega)$ given and $t>0$, we
obtain that

$0\leq\langle\eta,w\rangle-\left\langle-\Delta_{p}(u-tw)-\mu\Delta_{q}(u-tw),w\right\rangle$

taking $t\rightarrow 0^{+}$ gives that

$0\leq\langle\eta,w\rangle-\left\langle-\Delta_{p}(u)-\mu\Delta_{q}(u),w\right\rangle.$

Thus (25) deduced and this completes the proof. ∎

Focusing on the fractional parameter we can redefine our equation (2) as
follows

(26) $\left\\{\begin{aligned}
(-\Delta)_{p}^{s}\,u+\mu(-\Delta)_{q}^{s}\,u+|u|^{p-2}u+\mu|u|^{q-2}u=\lambda^{s}\
V(x)|u|^{p-2}u\quad&\text{in }\Omega\\\
u=0\quad&\text{in}\quad\mathbb{R}^{N}\backslash\Omega,\end{aligned}\right.$

and its local counterpart

(27) $\left\\{\begin{aligned}
(-\Delta)_{p}\,u+\mu(-\Delta)_{q}\,u+|u|^{p-2}u+\mu|u|^{q-2}u=\lambda^{1}\
V(x)|u|^{p-2}u\quad&\text{in }\Omega\\\
u=0\quad&\text{in}\quad\mathbb{R}^{N}\backslash\Omega.\end{aligned}\right.$

######  Theorem 5.2.

Let $\lambda^{s}\in\sigma_{p,q}^{s}$ be such that
$\sup_{s\in(0,1)}\lambda^{s}<\infty.$ Then any accumulation point of the set
$\\{\lambda^{s}\\}_{s\in(0,1)}$ belongs to $\sigma^{1}_{p,q}.$ Moreover, if
$\left\\{s_{k}\right\\}_{k\in\mathbb{N}}$ is such that $s_{k}\rightarrow 1$
and $\lambda^{s_{k}}\rightarrow\lambda$ and $u_{k}\in\mathcal{W}^{s_{k}}$ is
an eigenfunction of (26) associated to $\lambda^{s_{k}}$, then, up to a
further subsequence, there exists $u\in W_{0}^{1,p}(\Omega)$ such that
$u_{k}\rightarrow$ u strongly in $L^{p}(\Omega)$ and $u$ is an eigenfunction
of (27) associated to $\lambda$.

###### Proof.

Assume that $\lambda^{k}:=\lambda^{s_{k}}\rightarrow\lambda>\lambda_{1}(p,V)$
and let $u_{k}$ be the associated eigenfunction of (26). First, let us prove
that

(28) $\left\|u_{k}\right\|_{s_{k},p}^{p}\leq\|u_{k}\|_{s_{k}}\leq M,$

with $M$ independent on $k\in\mathbb{N}$ and
$\|\cdot\|_{s}:=\|\cdot\|_{s,p}^{p}+\|\cdot\|_{s,q}^{q}.$

Since $\lambda^{k}\searrow\lambda>\lambda_{1}(p,V)$, then for some $s\in(0,1)$
and $0<\varepsilon<\lambda,$ there exists
$\omega\in\mathcal{W}^{s}\backslash\\{0\\}$ (independent of $k$) for which

(29)
$\|\omega\|_{s,p}^{p}<(\lambda-\varepsilon)\|\omega\|_{p,V}^{p}<\lambda^{k}\|\omega\|_{p,V}^{p}.$

Consider the function $\Phi\colon(0,+\infty)\rightarrow\mathbb{R}$ defined by

$\displaystyle\Phi(t)=\mathcal{J}(t\omega)$
$\displaystyle=t^{p}\frac{\|\omega\|_{s,p}^{p}}{p}+t^{q}\frac{\|\omega\|_{s,q}^{q}}{q}-t^{p}\lambda^{k}\frac{\|\omega\|_{p,V}^{p}}{p}$
$\displaystyle:=\alpha t^{p}+\beta
t^{q}-\gamma_{k}t^{p}=(\alpha-\gamma_{k})t^{p}+\beta t^{q}.$

Passing to the derivative:

$\Phi^{\prime}(t)=0\quad\Leftrightarrow\quad t_{k}=\left(\frac{\beta
q}{p(\alpha-\gamma_{k})}\right)^{\frac{1}{p-q}}.$

From (29), we get that $t_{k}\rightarrow t_{0}$ for some $t_{0}>0$, then
$\Phi^{\prime}(t_{0})=0$ and

(30) $e_{0}:=t_{0}\omega\in\mathcal{N}_{\lambda_{k}}\quad\text{for some}\
k_{0},\ \text{and all}\ k>k_{0}.$

Note that for each $u_{k}\in\mathcal{N}_{\lambda_{k}}$ the functional
$\mathcal{J}$ has the following expression

(31)
$\mathcal{J}(u_{k})=\mu\left(\frac{1}{q}-\frac{1}{p}\right)\|u_{k}\|_{s,q}^{q}\geq
0.$

On the other hand, from (30) we have
$\mathcal{J}(u_{k})=m_{\lambda_{k}}\leq\mathcal{J}(e_{0}).$ So

(32) $|\mathcal{J}(u_{k})|\leq c,\quad\text{for some}\ c>0.$

Let $\theta>p>q$. We can see that

$\displaystyle C_{0}(1+\|u_{k}\|_{s_{k}})$
$\displaystyle\geq\mathcal{J}(u_{k})+\frac{1}{\theta}\left\langle\mathcal{J}^{\prime}(u_{k}),u_{k}\right\rangle$
$\displaystyle=\left(\frac{1}{p}-\frac{1}{\theta}\right)\|u_{k}\|_{s,p}^{p}+\left(\frac{1}{q}-\frac{1}{\theta}\right)\|u_{k}\|_{s,q}^{q}-\left(\frac{1}{p}-\frac{1}{\theta}\right)\lambda^{k}\|u_{k}\|_{p,V}^{p}$
$\displaystyle\geq\left(\frac{1}{p}-\frac{1}{\theta}\right)\left(\|u_{k}\|_{s,p}^{p}+\|u_{k}\|_{s,q}^{q}\right)=:C_{1}\left(\|u_{k}\|_{s,p}^{p}+\|u_{k}\|_{s,q}^{q}\right).$

Now, assume by contradiction that $\|u_{k}\|_{s_{k}}\rightarrow\infty.$ Then
from (31) and (32), we have $\|u_{k}\|_{s_{k},p}^{p}\rightarrow\infty$ and
$\|u_{k}\|_{s_{k},q}^{q}$ is bounded. Therefore,

$C_{0}\left(\|u_{k}\|_{s_{k},p}+\|u_{k}\|_{s_{k},q}\right)=C_{0}(1+\|u_{k}\|_{s_{k}})\geq
C_{2}\|u_{k}\|_{s_{k},p}^{p},$

and thus

$C_{0}\left(\frac{1}{\|u_{k}\|_{s,p}^{p}}+\frac{1}{\|u_{k}\|_{s,p}^{p-1}}+\frac{1}{\|u_{k}\|_{s,p}^{p-q}}\right)\geq
C_{2}.$

Since $p>q>1$ and passing to the limit as $k\rightarrow\infty$ we deduce that
$C_{2}\leq 0$ which is a contradiction.

Therefore, (28) is proved and by [17], there exists $u\in W_{0}^{1,p}(\Omega)$
such that $u_{k}\rightarrow u$ strongly in $L^{p}(\Omega)$.

We claim that $u\neq 0,$ otherwise, we have
$\left\|u_{k}\right\|_{s_{k},p}\rightarrow 0,$ and
$\left\|u_{k}\right\|_{V,p}^{p}\rightarrow 0.$ By arguing as the proof of
Lemma 4.2 $(i)$, we get that

$\omega_{n}\rightharpoonup\omega\ \text{in}\
\mathcal{W}^{s}\quad\text{and}\quad\omega_{n}\rightarrow\omega\ \text{in}\
L^{p}(V,\Omega),$

with $\omega\neq 0.$ Fix $\varphi\in C^{\infty}_{0}$ such that

$\left\langle\left(-\Delta\right)_{q}\omega,\varphi\right\rangle+\int_{\Omega}|\omega|^{q-2}\omega\varphi
dx>0.$

Then, by (26), we have

$\left\langle\left(-\Delta_{p}\right)^{s_{k}}\omega_{k},\varphi\right\rangle+\int_{\Omega}|\omega_{k}|^{p-2}\omega_{k}\varphi\,dx+\frac{\mu}{\left\|u_{k}\right\|_{V,p}^{p-q}}\left(\left\langle\left(-\Delta_{q}\right)^{s_{k}}\omega_{k},\varphi\right\rangle+\int_{\Omega}|\omega_{k}|^{q-2}\omega_{k}\varphi\,dx\right)=\lambda^{s_{k}}.$

The left-hand side of the previous equality tends towards infinity on the
other hand the term on the right is bounded, which is a contradiction, so we
have $u\neq 0.$

Now, since $f_{k}(z):=\lambda^{k}V|z|^{p-2}z\rightarrow f(z):=\lambda
V|z|^{p-2}z$ uniformly on compact sets of $z\in\mathbb{R}$, from Theorem 5.1
it follows that $u$ is an eigenfunction of (27) associated to $\lambda$ as we
wanted to show. ∎

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