# Fractional double phase Robin problem involving variable order-exponents
without Ambrosetti-Rabinowitz condition

Reshmi Biswas  ,  Sabri Bahrouni  and  Marcos L. Carvalho  Mathematics
Department, Indian Institute of technology Guwahati, Guwahati, Assam 781039,
India  Mathematics Department, Faculty of Sciences, University of Monastir,
5019 Monastir, Tunisia  Mathematics Institute, Universidade Federal de Goiás,
Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider a fractional double phase Robin problem involving variable order
and variable exponents. The nonlinearity $f$ is a Carathéodory function
satisfying some hypotheses which do not include the Ambrosetti-Rabinowitz type
condition. By using a Variational methods, we investigate the multiplicity of
solutions.

###### Key words and phrases:

Variable-order fractional $p(\cdot)$-Laplacian, Double phase problem, Robin
boundary condition, variational methods
2010 Mathematics Subject Classifications: 35R11, 35S15, 47G20, 47J30.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries results
    1. 2.1 Variable exponent Lebesgue spaces
    2. 2.2 Variable order fractional Sobolev spaces with variable exponents
  3. 3 Functional setting
  4. 4 Proof of Theorem 1.1
    1. 4.1 Abstract results
    2. 4.2 Geometric condition
    3. 4.3 Cerami condition
  5. 5 Proof of Theorem 1.2
  6. 6 Proof of Theorem 1.3
  7. 7 Proof of Theorem 1.4
  8. 8 Proof of Theorem 1.5

##  1\. Introduction

In the last few decades, problems involving $p(x)$-Laplacian, defined as
$(-\Delta)_{p(x)}u:=(|\nabla u|^{p(x)-2}\nabla u),\ x\in\mathbb{R}^{N},$ were
$p:\mathbb{R}^{N}\to[1,\infty)$ is continuous function, have been studied
intensively due to its major real world appearances in several mathematical
models, for e.g., electrorheological fluid flow, image restorations, etc. (see
[1, 14, 42, 49]). Various parametric boundary value problems with variable
exponents can be found in the book of Rădulescu-Repovš [39] and also one can
refer to the book by Diening et al. [17] for the properties of such operator
and associated variable exponent Lebesgue spaces and variable exponent Sobolev
spaces.
On the other hand, recently, great attention has been focused on the study of
fractional and nonlocal operators of elliptic type, both for pure mathematical
research and in view of concrete real-world applications. In most of these
applications, a fundamental tool to treat these type of problems is the so-
called fractional order Sobolev spaces. The literature on nonlocal operators
and on their applications is very interesting and, up to now, quite large. We
also refer to the recent monographs [18, 36] for a thorough variational
approach of nonlocal problems.

A bridge between fractional order theories and Sobolev spaces with variable
settings is first provided in [27]. In that paper, the authors defined the
Fractional Sobolev spaces with variable exponents and introduce the
corresponding fractional $p(\cdot)$-Laplacian as

(1.1) $(-\Delta)_{p(\cdot)}^{s}u(x):=P.V.\int_{\Omega}\frac{\mid
u(x)-u(y)\mid^{p(x,y)-2}(u(x)-u(y))}{\mid
x-y\mid^{N+sp(x,y)}}dy,~{}~{}x\in\Omega,$

where P.V. denotes Cauchy’s principal value,
$p:\Omega\times\Omega\to\mathbb{R}$ is a continuous function with
$1<p(x,y)<\infty$ and $0<s<1$, where $\Omega$ is a smooth domain. The idea of
studying such spaces and the associated operator defined in (1.1) arises from
a natural inquisitiveness to see what results can be recovered when the
standard local $p(x)$-Laplace operator is replaced by the fractional
$p(\cdot)$-Laplacian. Continuing with this thought and inspired by the vast
applications of variable order derivative (see for e.g. [29, 30, 32, 33, 41,
44] and references there in), Biswas and Tiwari [11] introduced the variable
order fractional Sobolev spaces with variable exponent and corresponding
variable-order fractional $p(\cdot)$-Laplacian by imposing variable growth on
the fractional order $s$, given in (1.1), to study some elliptic problems. In
fact, results regarding fractional $p(\cdot)$-Laplace equations and variable-
order fractional $p(\cdot)$-Laplace equations are in progress, for example, we
refer to [4, 9, 24, 25] and [6, 12, 47], respectively.

In this paper, we are interested in the following problem:

(1.2)
$\begin{cases}\mathcal{L}_{p_{1},p_{2}}^{s}(u)+|u|^{\overline{p}_{1}(x)-2}u+|u|^{\overline{p}_{2}(x)-2}u=f(x,u)&\mbox{in}\
\Omega\\\
\mathcal{N}^{s}_{p_{1},p_{2}}(u)+\beta(x)(|u|^{\overline{p}_{1}(x)-2}u+|u|^{\overline{p}_{2}(x)-2}u)=0&\mbox{in}\
\mathbb{R}^{N}\setminus\overline{\Omega},\end{cases}$

where

$\mathcal{L}_{p_{1},p_{2}}^{s}(u):=(-\Delta)^{s(.)}_{p_{1}(.)}(u)+(-\Delta)^{s(.)}_{p_{2}(.)}(u),$
$\mathcal{N}^{s}_{p_{1},p_{2}}(u):=\mathcal{N}^{s(\cdot)}_{p_{1}(\cdot)}(u)+\mathcal{N}^{s(\cdot)}_{p_{2}(\cdot)}(u)$

and

(1.3) $\displaystyle(-\Delta)^{s(.)}_{p_{i}(.)}u(x)=\displaystyle
P.V.\int_{\Omega}\frac{|u(x)-u(y)|^{p_{i}(x,y)}}{|x-y|^{N+s(x,y)p_{i}(x,y)}}\,dy,\;i=1,2,\quad\text{for
}x\in{\Omega},$ (1.4)
$\displaystyle\mathcal{N}^{s(\cdot)}_{p_{i}(\cdot)}u(x)=\int_{\Omega}\frac{|u(x)-u(y)|^{p_{i}(x,y)-2}(u(x)-u(y))}{|x-y|^{N+s(x,y)p_{i}(x,y)}}\,dy\quad\text{for
}x\in\mathbb{R}^{N}\setminus\overline{\Omega}.$

Here P.V. denotes the Cauchy’s principal value, $\Omega\subset\mathbb{R}^{N}$
is a bounded smooth domain, $s,p_{1},p_{2}$ are continuous functions such that
$\overline{p}_{i}(x):=p_{i}(x,x),~{}i=1,2,~{}\overline{s}(x):=s(x,x)$ with
appropriate assumptions described later. The variable exponent $\beta$
verifies the assumption

($\beta$) $\beta\in
L^{\infty}(\mathbb{R}^{N}\setminus\Omega)\quad\text{and}\quad\beta\geq 0\
\text{in}\ \mathbb{R}^{N}\setminus\Omega.$

The operator, defined in (1.2), is called double phase type operator which has
some important applications in biophysics, plasma physics, reaction-diffusion,
etc. (see [13, 22, 45], for e.g.). For more details on applications of such
operators in constant exponent set up, that is , $(p,q)$-Laplacian equations,
we refer to the survey article [34], see also [8, 40] for the nonconstant
case. This present paper generalizes some results contained in [37] and [43]
to the case of nonlocal partial differential equations with variable
exponents. If $p_{1}$ and $p_{2}$ are constants, then (1.2) becomes the usual
nonlocal constant exponent differential equation discussed in [3, 15]. Several
results for $(p,q)$-Laplacian problems set in bounded domains and in the whole
of $R^{N}$ can be found in [10, 21, 35] and the references therein. But if
either $p_{1}$ or $p_{2}$ is a non-constant function, then (1.2) has a more
complicated structure, due to its non-homogeneity and to the presence of
several nonlinear terms, only few recent works deal with these problems. For
instance, in [48], the authors generalize the double phase problem involving a
local version of the fractional operator with variable exponents, discussed in
[16], and studied the problem involving variable order fractional
$p(\cdot)\&q(\cdot)$-Laplacian but with homogeneous Dirichlet boundary datum,
that is, $u=0$ in $\mathbb{R}^{N}\setminus\Omega.$

Now we consider some notations as follows. For any set $\mathcal{D}$ and any
function $\Phi:\mathcal{D}\rightarrow\mathbb{R}$, we fix

$\displaystyle\Phi^{-}:=\inf_{\mathcal{D}}\Phi(x)\text{ ~{}~{}~{}and
~{}~{}}\Phi^{+}:=\sup_{\mathcal{D}}\Phi(x).$

We define the function space

$C_{+}(\mathcal{D}):=\\{\Phi:\mathcal{D}\to\mathbb{R}\text{~{}is uniformly
continuous~{}}:~{}1<\Phi^{-}\leq\Phi^{+}<\infty\\}.$

We consider the following hypotheses on the variable order $s$ and on the
variable exponents $p_{1},p_{2}:$

  * $(H_{1})$

$s:\mathbb{R}^{N}\times\mathbb{R}^{N}\rightarrow(0,1)$ is a uniformly
continuous and symmetric function, i.e., $s(x,y)=s(y,x)$ for all
$(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$ with $0<s^{-}\leq s^{+}<1$.

  * $(H_{2})$

$p_{i}\in C_{+}(\mathbb{R}^{N}\times\mathbb{R}^{N})$ are uniformly continuous
and symmetric functions, i.e., $p_{i}(x,y)=p_{i}(y,x),i=1,2$ for all
$(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$ with $1<p_{1}^{-}\leq
p_{1}^{+}<p_{2}^{-}\leq p_{2}^{+}<+\infty$ such that $s^{+}p_{i}^{+}<N$.

First we study our problem without assuming the well known Ambrosetti-
Rabinowitz (AR, in short) type condition on the nonlinearity $f$, which is
given as

$(AR)\qquad\quad{\exists\theta>p_{2}^{+}}\text{ s.t.}\;\;tf(x,t)>\theta
F(x,t),\;\;\forall|t|>0.$

As known, under $(AR)$, any Palais-Smale sequence of the corresponding energy
functional is bounded, which plays an important role of the application of
variational methods. In our problem the nonlinearity
$f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function
such that $f(x,0)=0$ for a.e. $x\in\Omega.$ The further assumptions on $f$ are
given below.

  * $(f_{1})$

There exists $a\in L^{\infty}(\Omega)$ such that $|f(x,t)|\leq
a(x)\left(1+|t|^{r(x)-1}\right),$ for a.e. $x\in\Omega$ and for all
$t\in\mathbb{R},$ where $r\in C_{+}(\mathbb{R}^{N})$ with $p_{2}^{+}<r^{-}\leq
r(x)<\frac{N\overline{p}_{2}(x)}{N-\overline{s}(x)\overline{p}_{2}(x)}:={{p_{2}}_{s}}^{*}(x)$.

  * $(f_{2})$

If $F(x,t):=\int_{0}^{t}f(x,s)ds$, then
$\displaystyle\lim_{|t|\to+\infty}\frac{F(x,t)}{|t|^{p_{2}^{+}}}=0$ uniformly
for $a.e$ $x\in\Omega.$

  * $(f_{3})$

$\displaystyle\lim_{|t|\to 0}\frac{f(x,t)}{|t|^{p_{2}^{+}-2}t}=0$ uniformly
for $a.e$ $x\in\Omega$.

  * $(f_{4})$

Let $\mathcal{F}(x,t)=tf(x,t)-p_{2}^{+}F(x,t).$ Then there exists $b\in
L^{1}(\Omega)$ such that

$\mathcal{F}(x,t)\leq\mathcal{F}(x,\tau)+b(x)\ \text{for a.e.}\ x\in\Omega,\
\text{all}\ 0\leq t\leq\tau\ \text{or all}\ \tau\leq t\leq 0.$

Consider the following function

$g(x,t)=t|t|^{{\frac{p_{2}^{+}}{2}}-2}\log(1+|t|).$

One can check that $g$ does not satisfy $(AR)$ but it satisfies
$(f_{1})$-$(f_{4})$. Therefore by dropping $(AR)$ condition, not only we
invite complications in the compactness of Palais-Smale sequence but also we
include larger class of nonlinearities. To overcome such aforementioned
difficulty, we analyze the Cerami condition (see Definition 4.1), which is
more appropriate for the set up of our problem. Finally, we are in a position
to state the main results of this article.

######  Theorem 1.1.

Let hypotheses $(H_{1})$-$(H_{2})$, $(\beta)$ and $(f_{1})$-$(f_{4})$ hold.
Then there exists a non-trivial weak solution of (1.2).

Next, for the odd nonlinearity $f(x,t),$ we state the existence results of
infinitely many solutions using the Fountain theorem and the Dual fountain
theorem, respectively.

######  Theorem 1.2.

Let hypotheses $(H_{1})$-$(H_{2})$, $(\beta)$ and $(f_{1})$-$(f_{4})$ hold.
Also let $f(x,-t)=-f(x,t).$ Then the problem (1.2) has a sequence of
nontrivial weak solutions with unbounded energy.

######  Theorem 1.3.

Let hypotheses $(H_{1})$-$(H_{2})$, $(\beta)$ and $(f_{1})$-$(f_{4})$ hold.
Also let $f(x,-t)=-f(x,t)$. Then the problem (1.2) has a sequence of
nontrivial weak solutions with negative critical values converging to zero.

We prove the next theorem using the symmetric mountain pass theorem.

######  Theorem 1.4.

Let hypotheses $(H_{1})$-$(H_{2})$, $(\beta)$ and $(f_{1})$-$(f_{4})$ hold.
Also let $f(x,-t)=-f(x,t)$. Then the problem (1.2) has a sequence of
nontrivial weak solutions with unbounded energy characterized by a minmax
argument.

In the next theorem, we consider the following concave and convex type
nonlinearity $f:$

  * $(f_{5})$

For $\lambda>0$ and $q,r\in C_{+}(\Omega)$ with $1<q^{-}\leq q^{+}<p_{1}^{-}$
and $p_{2}^{+}<r^{-}$

$f(x,t)=\lambda|t|^{q(x)-2}t+|t|^{r(x)-2}t$

######  Theorem 1.5.

Let hypotheses $(H_{1})$-$(H_{2})$, $(\beta)$ and $(f_{5})$ hold. Then for all
$\lambda>0,$ the problem (1.2) has a sequence of nontrivial weak solutions
converging to $0$ with negative energy.

It is noteworthy to mention that we are the first ( as per the best of our
knowledge) to study the above existence results for the problem (1.2) driven
by double phase variable-order fractional
$p_{1}(\cdot)\&p_{2}(\cdot)$-Laplacian involving Robin boundary condition and
non-AR type nonlinearities.

######  Remark 1.6.

Throughout this paper $C$ represents generic positive constant which may vary
from line to line.

##  2\. Preliminaries results

###  2.1. Variable exponent Lebesgue spaces

In this section first we recall some basic properties of the variable exponent
Lebesgue spaces, which we will use to prove our main results.

For $q\in C_{+}(\Omega)$ define the variable exponent Lebesgue space
$L^{q(\cdot)}(\Omega)$ as

$L^{q(\cdot)}(\Omega):={\Big{\\{}u:\Omega\to\mathbb{R}\
\text{is~{}measurable}:\int_{\Omega}|u(x)|^{q(x)}\;dx<+\infty\Big{\\}}}$

which is a separable, reflexive, uniformly convex Banach space (see [17, 19])
with respect to the Luxemburg norm

$\|{u}\|_{L^{q(\cdot)}(\Omega)}:=\inf\Big{\\{}\eta>0:\int_{\Omega}\Big{|}\frac{u(x)}{\eta}\Big{|}^{q(x)}\;dx\leq
1\Big{\\}}.$

Define the modular $\rho_{\Omega}^{q}:\ L^{q(\cdot)}(\Omega)\to\mathbb{R}$ as

$\rho_{\Omega}^{q}(u):=\int_{\Omega}|u|^{q(x)}\;dx,\ {for~{}all~{}}u\in
L^{q(\cdot)}(\Omega).$

######  Proposition 2.1.

([19])Let $u_{n},u\in L^{q(\cdot)}(\Omega)\setminus\\{0\\},$ then the
following properties hold:

  * (i)

$\eta=\|u\|_{L^{q(\cdot)}(\Omega)}$ if and only if
$\rho_{\Omega}^{q}(\frac{u}{\eta})=1.$

  * (ii)

$\rho_{\Omega}^{q}(u)>1$ $(=1;\ <1)$ if and only if
$\|u\|_{L^{q(\cdot)}(\Omega)}>1$ $(=1;\ <1)$, respectively.

  * (iii)

If $\|u\|_{L^{q(\cdot)}(\Omega)}>1$, then
$\|u\|_{L^{q(\cdot)}(\Omega)}^{p^{-}}\leq\rho_{\Omega}^{q}(u)\leq\|u\|_{L^{q(\cdot)}(\Omega)}^{p^{+}}$.

  * (iv)

If $\|u\|_{L^{q(\cdot)}(\Omega)}<1$, then
$\|u\|_{L^{q(\cdot)}(\Omega)}^{p^{+}}\leq\rho_{\Omega}^{q}(u)\leq\|u\|_{L^{q(\cdot)}(\Omega)}^{p^{-}}$.

  * (v)

${\displaystyle\lim_{n\to+\infty}}\|u_{n}-u\|_{L^{q(\cdot)}(\Omega)}=0\iff{\displaystyle\lim_{n\to+\infty}}\rho_{\Omega}^{q}(u_{n}-u)=0.$

Let $q^{\prime}$ be the conjugate function of $q,$ that is,
$1/q(x)+1/q^{\prime}(x)=1$.

######  Proposition 2.2.

(Hölder inequality) ([19]) For any $u\in L^{q(\cdot)}(\Omega)$ and $v\in
L^{q^{\prime}(\cdot)}(\Omega)$, we have

$\Big{|}\int_{\Omega}uv\,dx\Big{|}\leq
2\|u\|_{L^{q(\cdot)}(\Omega)}\|{v}\|_{L^{q^{\prime}(\cdot)}(\Omega)}.$

######  Lemma 2.3.

([23, Lemma A.1]) Let $\vartheta_{1}(x)\in L^{\infty}(\Omega)$ such that
$\vartheta_{1}\geq 0,\;\vartheta_{1}\not\equiv 0.$ Let
$\vartheta_{2}:\Omega\to\mathbb{R}$ be a measurable function such that
$\vartheta_{1}(x)\vartheta_{2}(x)\geq 1$ a.e. in $\Omega.$ Then for every
$u\in L^{\vartheta_{1}(x)\vartheta_{2}(x)}(\Omega),$

$\parallel|u|^{\vartheta_{1}(\cdot)}\parallel_{L^{\vartheta_{2}(x)}(\Omega)}\leq\parallel
u\parallel_{L^{\vartheta_{1}(x)\vartheta_{2}(x)}(\Omega)}^{\vartheta_{1}^{-}}+\parallel
u\parallel_{L^{\vartheta_{1}(x)\vartheta_{2}(x)}(\Omega)}^{\vartheta_{1}^{+}}.$

###  2.2. Variable order fractional Sobolev spaces with variable exponents

Next, we define the fractional Sobolev spaces with variable order and variable
exponents (see [11]). Define

$\displaystyle W$
$\displaystyle=W^{s(\cdot,\cdot),\overline{p}(\cdot),p(\cdot,\cdot)}(\Omega)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}:=\Big{\\{}u\in
L^{\overline{p}(\cdot)}(\Omega):\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{\eta^{p(x,y)}|x-y|^{N+s(x,y)p(x,y)}}dxdy<\infty,\text{
for some }\eta>0\Big{\\}}$

endowed with the norm

$\|u\|_{W}:=\inf\Big{\\{}\eta>0:\rho_{W}\left(\frac{u}{\eta}\right)<1\Big{\\}},$

where

$\rho_{W}(u):=\int_{\Omega}\left|u\right|^{\overline{p}(x)}dx+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+s(x,y)p(x,y)}}\,dxdy$

is a modular on $W.$ Then, $(W,\|\cdot\|_{W})$ is a separable reflexive Banach
space (see [11, 24]). On $W$ we also make use of the following norm:

$|u|_{W}:=\|u\|_{L^{\overline{p}(\cdot)}(\mathbb{R}^{N})}+[u]_{W},$

where the seminorm $[\cdot]_{W}$ is defined as follows:

$[u]_{W}:=\inf\Big{\\{}\eta>0:\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{\eta^{p(x,y)}|x-y|^{N+s(x,y)p(x,y)}}\,dxdy<1\Big{\\}}.$

Note that $\|\cdot\|_{W}$ and $|\cdot|_{W}$ are equivalent norms on $W$ with
the relation

(2.1) $\frac{1}{2}\|u\|_{W}\leq|u|_{W}\leq 2\|u\|_{W},\ \
\text{~{}for~{}all~{}}u\in W.$

The following embedding result is studied in [11]. We also refer to [24] where
the authors proved the same result when $s(x,y)=s,$ constant.

######  Theorem 2.4 (Sub-critical embedding).

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ or
$\Omega=\mathbb{R}^{N}$. Let $s$ and $p$ satisfy $(H_{1})$ and $(H_{2}),$
respectively and $\gamma\in C_{+}(\overline{\Omega})$ satisfy
$1<\gamma(x)<p_{s}^{*}(x)$ for all $x\in\overline{\Omega}$. In addition, when
$\Omega=\mathbb{R}^{N}$, $\gamma$ is uniformly continuous and
$\overline{p}(x)<\gamma(x)$ for all $x\in\mathbb{R}^{N}$ and
$\inf_{x\in\mathbb{R}^{N}}\left(p_{s}^{*}(x)-\gamma(x)\right)>0$. Then, it
holds that

(2.2) $W\hookrightarrow L^{\gamma(\cdot)}(\Omega).$

Moreover, the embedding is compact.

Notations:

  * •

$\delta_{\Omega}^{p}(u)=\displaystyle\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+s(x,y)p(x,y)}}\,dxdy.$

  * •

For any measurable set $\mathcal{S},$ $|\mathcal{S}|$ denotes the Lebesgue
measure of the set.

##  3\. Functional setting

Now, we give the variational framework of problem (1.2). Let $s,p$ satisfy
$(H_{1})$, $(H_{2}),$ respectively. We set

$\displaystyle|u|_{{X{{}_{p}}}}:=[u]_{s(\cdot),p(\cdot),\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}+\|u\|_{L^{\overline{p}(\cdot)}(\Omega)}+\left\|\beta^{\frac{1}{\overline{p}(\cdot)}}u\right\|_{L^{\overline{p}(\cdot)}(\mathcal{C}\Omega)},$

where $\mathcal{C}\Omega=\mathbb{R}^{N}\setminus\Omega$ and

$\displaystyle
X^{s(\cdot)}_{p(\cdot)}:=\left\\{u\colon\mathbb{R}^{N}\to\mathbb{R}\text{
measurable }:\ \|u\|_{{X}_{p}}<\infty\right\\}.$

By following standard arguments, it can be seen that $X^{s(\cdot)}_{p(\cdot)}$
is reflexive Banach space with respect to the norm $|\cdot|_{X_{p}}$ (see [7,
Proposition 3.1]).

Note that the norm $|\cdot|_{X_{p}}$ is equivalent on
$X_{p(\cdot)}^{s(\cdot)}$ to the following norm:

(3.1) $\displaystyle\begin{split}\|u\|_{X_{p}}&=\inf\left\\{\eta\geq 0\
\bigg{|}\ \rho_{p}\left(\frac{u}{\eta}\right)\leq 1\right\\}\\\
&=\inf\left\\{\eta\geq 0\ \bigg{|}\
\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(u)|^{p(x,y)}}{\eta^{p(x,y)}p(x,y)|x-y|^{N+s(x,y)p(x,y)}}\,dx\,dy+\int_{\Omega}\frac{|u|^{\overline{p}(x)}}{\overline{p}(x)\eta^{\overline{p}(x)}}\,dx\right.\\\
&\qquad\qquad\qquad\quad\left.+\int_{\mathcal{C}\Omega}\frac{\beta(x)}{\eta^{\overline{p}(x)}\overline{p}(x)}|u|^{\overline{p}(x)}\,dx\leq
1\right\\},\end{split}$

where the modular $\rho_{p}\colon X_{p(\cdot)}^{s(\cdot)}\to\mathbb{R}$ is
defined by

$\displaystyle\rho_{p}\left(u\right)$
$\displaystyle=\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(u)|^{p(x,y)}}{p(x,y)|x-y|^{N+s(x,y)p(x,y)}}\,dx\,dy+\int_{\Omega}\frac{|u|^{\overline{p}(x)}}{\overline{p}(x)}\,dx$
(3.2)
$\displaystyle\quad+\int_{\mathcal{C}\Omega}\frac{\beta(x)}{\overline{p}(x)}|u|^{\overline{p}(x)}\,dx.$

The following lemma will be helpful in later considerations. The proof of this
lemma follows using the similar arguments as in [19].

######  Lemma 3.1.

Let $s,p$ and $\beta$ satisfy $(H_{1})$, $(H_{2})$ and $(\beta),$
respectively, and let $u\in X_{p(\cdot)}^{s(\cdot)}$. Then the following hold:

  1. (i)

For $u\neq 0$ we have: $\|u\|_{X_{p}}=\eta$ if and only if
$\rho_{p}(\frac{u}{\eta})=1$;

  2. (ii)

If $\|u\|_{X_{p}}<1$ then
${\|u\|_{X_{p}}^{p{+}}}\leq\rho_{p}(u)\leq\|u\|_{X_{p}}^{p^{-}}$;

  3. (iii)

If $\|u\|_{X_{p}}>1$ then
$\|u\|_{X_{p}}^{p^{-}}\leq\rho_{p}(u)\leq\|u\|_{X_{p}}^{p^{+}}$.

######  Lemma 3.2.

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}.$ Let $s$ and $p$
satisfy $(H1)$ and $(H2),$ respectively and $(\beta)$ hold. Then for any
$\gamma\in C_{+}(\overline{\Omega})$ satisfying $1<\gamma(x)<p_{s}^{*}(x)$ for
all $x\in\overline{\Omega},$ there exists a constant
$C(s,p,N,\gamma,\Omega)>0$ such that

$\|u\|_{L^{\gamma(\cdot)}(\Omega)}\leq
C(s,p,N,\gamma,\Omega)\|u\|_{X_{p}}\text{~{}~{}~{}~{}for all~{}~{}}u\in X,$

moreover this embedding is compact.

###### Proof.

It can easily be seen that $\|u\|_{W}\leq\|u\|_{X_{p}}.$ Now by applying
Theorem 2.4 we get our desired result. ∎

In order to deal with fractional $p_{1}(\cdot)\&p_{2}(\cdot)$-Laplacian
problems, we consider the space

$X:=X^{s(\cdot)}_{p_{1}(\cdot)}\cap X^{s(\cdot)}_{p_{2}(\cdot)}$

endowed with the norm

$|u|_{X}=\|u\|_{{X}_{p_{1}}}+\|u\|_{{X}_{p_{2}}}.$

Clearly $X$ is reflexive and separable Banach space with respect to the above
norm. It is not difficult to see we can make use of another norm on $X$
equivalent to $|\cdot|_{X}$ given as

$\|u\|:=\|u\|_{X}=\inf\left\\{\eta\geq 0\ \bigg{|}\
\rho\left(\frac{u}{\eta}\right)\leq 1\right\\},$

where the modular $\rho:X\to\mathbb{R}$ is defined as

$\rho(u)=\rho_{p_{1}}(u)+\rho_{p_{2}}(u)$

such that $\rho_{p_{1}},\rho_{p_{2}}$ are described as in (3).

######  Lemma 3.3.

Let hypotheses $(H_{1})$-$(H_{2})$ and $(\beta)$ be satisfied and let $u\in
X$. Then the following hold:

  1. (i)

For $u\neq 0$ we have: $\|u\|=\eta$ if and only if $\rho(\frac{u}{\eta})=1$;

  2. (ii)

If $\|u\|<1$ then $\|u\|^{p_{2}^{+}}\leq\rho(u)\leq\|u\|^{p_{1}^{-}}$;

  3. (iii)

If $\|u\|>1$ then $\|u\|^{p_{1}^{-}}\leq\rho(u)\leq\|u\|^{p_{2}^{+}}$.

######  Lemma 3.4.

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}.$ Let $s$ and
$p_{i}$ satisfy $(H1)$ and $(H2),$ respectively for $i=1,2$ and $(\beta)$
hold. Then for any $\gamma\in C_{+}(\overline{\Omega})$ satisfying
$1<\gamma(x)<{p}_{2_{s}}^{*}(x)$ for all $x\in\overline{\Omega},$ there exists
a constant $C(s,p_{i},N,\gamma,\Omega)>0$ such that

$\|u\|_{L^{\gamma(\cdot)}(\Omega)}\leq
C(s,p_{i},N,\gamma,\Omega)\|u\|\text{~{}~{}~{}~{}for all~{}~{}}u\in X,$

moreover this embedding is compact.

###### Proof.

The proof directly follows from the definition of $\|u\|$ and Lemma 3.2. ∎

Throughout this article $X^{*}$ represents the topological dual of $X$.

######  Lemma 3.5.

Let hypotheses $(H_{1})$-$(H_{2})$ and $(\beta)$ be satisfied. Then
$\rho:X\to\mathbb{R}$ and $\rho^{\prime}:X\to X^{*}$ have the following
properties:

  * $(i)$

The function $\rho$ is of class $C^{1}(X,\mathbb{R})$ and $\rho^{\prime}:X\to
X^{*}$ is coercive, that is,

$\frac{\langle\rho^{\prime}(u),u\rangle}{\|u\|}\to+\infty\text{\;\;as\;}\|u\|\to+\infty.$

  * $(ii)$

$\rho^{\prime}$ is strictly monotone operator.

  * $(iii)$

$\rho^{\prime}$ is a mapping of type $(S_{+})$, that is, if
$u_{n}\rightharpoonup u$ in $X$ and
$\displaystyle\limsup_{n\to+\infty}\langle\rho^{\prime}(u_{n}),u_{n}-u\rangle\leq
0$, then $u_{n}\to u$ strongly in $X$.

###### Proof.

The proof of this result is similar to the proof of [9, Lemma 4.2], just
noticing that, the quantities
$\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}$ and $\Omega\times\Omega$
play a symmetrical role. ∎

As proved in [7, Proposition 3.6], the following integration by parts formula
arises naturally for $u\in C^{2}$ functions:

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}|u(x)-u(y)|^{p(x,y)-2}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+s(x,y)p(x,y)}}\,dx\,dy$
(3.3)
$\displaystyle=\int_{\Omega}v(-\Delta)^{s(\cdot)}_{p(\cdot)}u\,dx+\int_{\mathcal{C}\Omega}v\mathcal{N}^{s(\cdot)}_{p(\cdot)}\,dx.$

The previous integration by parts formula leads to the following definition:

######  Definition 3.6.

We say that $u\in X$ is a weak solution to (1.2) if for any $v\in X$ we have

$\displaystyle\mathcal{H}(u,v)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{1}(x,y)-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+s(x,y)p_{1}(x,y)}}dxdy+\int_{\Omega}|u|^{\overline{p}_{1}(x)-2}uvdx$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{2}(x,y)-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+s(x,y)p_{2}(x,y)}}dxdy+\int_{\Omega}|u|^{\overline{p}_{2}(x)-2}uvdx$
$\displaystyle-\int_{\Omega}f(x,u)v\,dx+\int_{\mathcal{C}\Omega}\beta(x)|u|^{\overline{p}_{1}(x)-2}uv\,dx+\int_{\mathcal{C}\Omega}\beta(x)|u|^{\overline{p}_{2}(x)-2}uv\,dx$
$\displaystyle=0.$

The problem taken into account in the present paper has a variational
structure, namely its solutions can be found as critical points of the
associated energy functional. Hence, our problem can be studied using all the
methods which aim to prove the existence of a critical point for a functional.

The energy functional associated with problem (1.2) is the functional
$\mathcal{I}\colon X\to\mathbb{R}$ given by

$\displaystyle\mathcal{I}(u)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{1}(x,y)}}{p_{1}(x,y)|x-y|^{N+s(x,y)p_{1}(x,y)}}dxdy+\int_{\Omega}\frac{1}{\overline{p}_{1}(x)}|u|^{\overline{p}_{1}(x)}dx$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{2}(x,y)}}{p_{2}(x,y)|x-y|^{N+s(x,y)p_{2}(x,y)}}dxdy+\int_{\Omega}\frac{1}{\overline{p}_{2}(x)}|u|^{\overline{p}_{2}(x)}dx$
$\displaystyle+\int_{\mathcal{C}\Omega}\frac{\beta(x)|u|^{\overline{p}_{1}(x)}}{\overline{p}_{1}(x)}\,dx+\int_{\mathcal{C}\Omega}\frac{\beta(x)|u|^{\overline{p}_{2}(x)}}{\overline{p}_{2}(x)}\,dx-\int_{\Omega}F(x,u)dx.$

A direct computation from [7, Proposition 3.8] shows that the functional
$\mathcal{I}$ is well defined on $X$ and $\mathcal{I}\in C^{1}(X,\mathbb{R})$
with

$\langle\mathcal{I}^{{}^{\prime}}(u),v\rangle=\mathcal{H}(u,v)\quad\text{for
any}\quad v\in X.$

Thus the weak solutions of (1.2) are precisely the critical points of
$\mathcal{I}$.

##  4\. Proof of Theorem 1.1

###  4.1. Abstract results

######  Definition 4.1.

Let $E$ be a Banach space and $E^{*}$ be its topological dual. Suppose that
$\Phi\in C^{1}(E)$. We say that $\Phi$ satisfies the Cerami condition at the
level $c\in\mathbb{R}$ $($the $(C)_{c}$-condition for short$)$ if the
following is true:

$``\text{every sequence}\ (u_{n})_{n\in\mathbb{N}}\subseteq E\ \text{such
that}\ \Phi(u_{n})\to c\ \text{and}$
$(1+\|u_{n}\|_{E})\Phi^{{}^{\prime}}(u_{n})\to 0\ \text{in}\ E^{*}\ \text{as}\
n\to+\infty$ $\text{admits a strongly convergent subsequence}".$

If this condition holds at every level $c\in\mathbb{R}$, then we say that
$\Phi$ satisfies the Cerami condition (the $C$-condition for short).

The $(C)_{c}$-condition is weaker than the $(PS)_{c}$-condition. However, it
was shown in [28] that from $(C)_{c}$-condition it can obtain a deformation
lemma, which is fundamental in order to get some minimax theorems. Thus we
have

######  Theorem 4.2.

If there exist $e\in E$ and $r>0$ such that

$\|e\|>r,\quad\max(\Phi(0),\Phi(e))\leq\inf_{\|x\|=r}\Phi(x),$

and $\Phi\colon E\to\mathbb{R}$ satisfies the $(C)_{c}$-condition with

$c=\inf_{\gamma\in\Gamma}\max_{t\in(0,1)}\Phi(\gamma(t)),$

where

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

Then $c\geq\inf_{\|x\|=r}\Phi(x)$ and $c$ is a critical value of $\Phi$.

###  4.2. Geometric condition

######  Lemma 4.3.

Let $(H_{1})$-$(H_{2})$, $(\beta)$ and $(f_{1})$-$(f_{4})$ hold. Then

  1. $(i)$

there exist $\alpha>0$ and $R>0$ such that

$\displaystyle\mathcal{I}(u)\geq\beta>0\quad\text{for any }u\in X\text{ with
}\ \|u\|=\alpha.$

  2. $(ii)$

there exists $\varphi\in X$ such that $I(\varphi)<0$.

###### Proof.

$(i)$ For any $\epsilon>0$, by the assumptions $(f_{1})$-$(f_{3})$, we have

(4.1) $F(x,t)\leq\epsilon|t|^{p_{2}^{+}}+C(\epsilon)|t|^{r(x)},\quad\text{for
a.e.}\ x\in\Omega\ \text{and all}\ t\in\mathbb{R}.$

Hence, using Theorem 2.4, Lemma 3.3, Lemma 2.3 and Lemma 3.4 for any $u\in X$
with $\|u\|<1$ $($ i.e. $\|u\|_{X_{p_{i}}}<1,i=1,2),$ we obtain

$\displaystyle\mathcal{I}(u)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{1}(x,y)}}{p_{1}(x,y)|x-y|^{N+s(x,y)p_{1}(x,y)}}dxdy+\int_{\Omega}\frac{1}{\overline{p}_{1}(x)}|u|^{\overline{p}_{1}(x)}dx$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{2}(x,y)}}{p_{2}(x,y)|x-y|^{N+s(x,y)p_{2}(x,y)}}dxdy+\int_{\Omega}\frac{1}{\overline{p}_{2}(x)}|u|^{\overline{p}_{2}(x)}dx$
$\displaystyle+\int_{\mathcal{C}\Omega}\frac{\beta(x)|u|^{\overline{p}_{1}(x)}}{\overline{p}_{1}(x)}\,dx+\int_{\mathcal{C}\Omega}\frac{\beta(x)|u|^{\overline{p}_{2}(x)}}{\overline{p}_{2}(x)}\,dx$
$\displaystyle\quad-\int_{\Omega}F(x,u)\,dx$
$\displaystyle\geq\frac{1}{2}\rho(u)-{\epsilon}\int_{\Omega}|u|^{p_{2}^{+}}\,dx-C(\epsilon)\int_{\Omega}|u|^{r(x)}\,dx$
$\displaystyle\geq\frac{1}{2}\|u\|^{p_{2}^{+}}-\epsilon\|u\|^{p_{2}^{+}}_{L^{p_{2}^{+}}(\Omega)}-C(\epsilon)\left\\{\|u\|^{r^{-}}_{L^{r(\cdot)}(\Omega)}+\|u\|^{r^{+}}_{L^{r(\cdot)}(\Omega)}\right\\}$
$\displaystyle\geq\frac{1}{2}\|u\|^{p_{2}^{+}}-\epsilon
C\|u\|^{p_{2}^{+}}-C^{{}^{\prime}}(\epsilon)\|u\|^{r^{-}}$
$\displaystyle=\left(\frac{1}{2}-\epsilon
C\right)\|u\|^{p_{2}^{+}}-C^{{}^{\prime}}(\epsilon)\|u\|^{r^{-}},$

where $C^{{}^{\prime}}(\epsilon)>0$ is a constant. Consider

$0<\epsilon<\frac{1}{4C}.$

Since $p_{2}^{+}<r^{-}$, we can choose $\alpha\in(0,1)$ sufficiently small
such that for all $u\in X$ with $\|u\|=\alpha$

$\displaystyle\mathcal{I}(u)\geq\alpha^{p_{2}^{+}}\left(\frac{1}{2}-{\epsilon
C}\right)-C^{{}^{\prime}}(\epsilon)\alpha^{r^{-}}=R>0.$

The proof of $(i)$ is complete.

$(ii)$ It follows from $(f_{1})$ and $(f_{2})$ that for any positive constant
$M$, there exists a corresponding positive constant $C_{M}$ such that

(4.2) $\displaystyle F(x,t)\geq{M}|t|^{p_{2}^{+}}-C_{M}.$

Let $e\in X,~{}e>0$ with $\|e\|=1$ and $\int_{\Omega}|e|^{p_{2}^{+}}dx>0$ and
$t>1$. Then, using Lemma 3.3 and (4.2), we get

$\displaystyle I(te)$
$\displaystyle=\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}t^{p_{1}(x,y)}\frac{|e(x)-e(y)|^{p_{1}(x,y)}}{2p_{1}(x,y)|x-y|^{N+sp_{1}(x,y)}}\,dx\,dy+\int_{\mathcal{C}\Omega}t^{\overline{p}_{1}(x)}\frac{\beta(x)|e|^{\overline{p}_{1}(x)}}{\overline{p}_{1}(x)}\,dx$
$\displaystyle\quad+\int_{\Omega}t^{\overline{p}_{1}(x)}\frac{|e|^{\overline{p}_{1}(x)}}{\overline{p}_{1}(x)}\,dx$
$\displaystyle+\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}t^{p_{2}(x,y)}\frac{|e(x)-e(y)|^{p_{2}(x,y)}}{2p_{2}(x,y)|x-y|^{N+sp_{2}(x,y)}}\,dx\,dy+\int_{\mathcal{C}\Omega}t^{\overline{p}_{2}(x)}\frac{\beta(x)|e|^{\overline{p}_{2}(x)}}{\overline{p}_{2}(x)}\,dx$
$\displaystyle\quad+\int_{\Omega}t^{\overline{p}_{2}(x)}\frac{|e|^{\overline{p}_{2}(x)}}{\overline{p}_{2}(x)}\,dx-\int_{\Omega}F(x,te)\,dx$
$\displaystyle\leq
t^{p_{2}^{+}}\rho(e)-t^{p_{2}^{+}}M\int_{\Omega}|e|^{p_{2}^{+}}\,dx+|\Omega|C_{M}$
$\displaystyle=t^{p_{2}^{+}}\left[1-M\int_{\Omega}|e|^{p_{2}^{+}}\,dx\right]+|\Omega|C_{M}$

We choose $M$ sufficiently large so that

$\lim_{t\to+\infty}\mathcal{I}(te)=-\infty.$

Hence, there exists some $t_{0}>0$ such that $\mathcal{I}(\varphi)<0,$ where
$\varphi=t_{0}e.$ Thus the proof of $(ii)$ is complete. ∎

###  4.3. Cerami condition

######  Proposition 4.4.

If hypotheses $(H_{1})$-$(H_{2})$, $(\beta)$ and $(f_{1})$-$(f_{4})$ hold,
then the functional $\mathcal{I}$ satisfies the $(C)_{c}$-condition for any
$c\in\mathbb{R}.$

###### Proof.

In this proof the value of the constant $C$ changes from line to line. We
consider a sequence $(u_{n})_{n\geq 1}\subset X$ such that

(4.3) $|\mathcal{I}(u_{n})|\leq C\quad\text{for some}\quad C>0\quad\text{and
for all}\ n\geq 1,$ (4.4) $(1+\|u_{n}\|)\mathcal{I}^{{}^{\prime}}(u_{n})\to
0\quad\text{in}\quad X^{*}\quad\text{as}\ n\to+\infty.$

From (4.4), we have

(4.5) $|\mathcal{H}(u_{n},v)|\leq\frac{\epsilon_{n}\|v\|}{1+\|u_{n}\|},$

for all $v\in X$ with $\epsilon_{n}\to 0$.

In (4.5), we choose $v=u_{n}\in X$ and obtain for all $n\in\mathbb{N}$

$\displaystyle-\frac{1}{2}\delta^{p_{1}}_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}(u_{n})-\frac{1}{2}\delta^{p_{2}}_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}(u_{n})-\rho_{\Omega}^{p_{1}}(u_{n})-\rho_{\Omega}^{p_{2}}(u_{n})$
$\displaystyle-\int_{C\Omega}\beta(x)|u_{n}|^{\overline{p}_{1}(x)}dx-\int_{C\Omega}\beta(x)|u_{n}|^{\overline{p}_{2}(x)}dx+\int_{\Omega}f(x,u_{n}(x))u_{n}(x)dx$
(4.6) $\displaystyle\leq\epsilon_{n}.$

Also, by (4.3) we have for all $n\in\mathbb{N}$,

$\displaystyle\frac{1}{2p_{1}^{+}}\delta^{p_{1}}_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}(u_{n})+\frac{1}{2p_{2}^{+}}\delta^{p_{2}}_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}(u_{n})+\frac{1}{p_{1}^{+}}\rho_{\Omega}^{p_{1}}(u_{n})+\frac{1}{p_{2}^{+}}\rho_{\Omega}^{p_{2}}(u_{n})$
$\displaystyle+\frac{1}{p_{1}^{+}}\int_{C\Omega}\beta(x)|u_{n}|^{\overline{p}_{1}(x)}dx+\frac{1}{p_{2}^{+}}\int_{C\Omega}\beta(x)|u_{n}|^{\overline{p}_{2}(x)}dx$
(4.7) $\displaystyle-\int_{\Omega}F(x,u_{n}(x))dx\leq C.$

Adding relations (4.3) and (4.3), we obtain

(4.8) $\int_{\Omega}\mathcal{F}(x,u_{n}(x))dx\leq C\quad\text{for some}\quad
C>0\quad\text{and all}\ n\in\mathbb{N}.$

Claim: The sequence $(u_{n})_{n\geq 1}\subset X$ is bounded.

We argue by contradiction. Suppose that the claim is not true. We may assume
that

(4.9) $\|u_{n}\|\to+\infty\quad\text{as}\quad n\to+\infty.$

We set $w_{n}:=\frac{u_{n}}{\|u_{n}\|}$ for all $n\in\mathbb{N}$. Then
$\|w_{n}\|=1$, for all $n\in\mathbb{N}$. Using reflexivity of $X$ and Lemma
3.4, up to a subsequence, still denoted by $(w_{n})_{n\geq 1}$, as
$n\to+\infty,$ we get

(4.10) $w_{n}\rightharpoonup w\ \text{~{}weakly in}\ X\quad\text{and}\quad
w_{n}\to w\ \text{~{} strongly in}\
L^{\gamma(\cdot)}(\Omega),\;1<\gamma(x)<{p_{2}}_{s}^{*}(x).$

We claim that $w=0$. Indeed, if not then the set
$\widehat{\Omega}:=\\{x\in\Omega:w(x)\not=0\\}$ has positive Lebesgue measure,
i.e., $|\widehat{\Omega}|>0.$ Hence, $|u_{n}(x)|\to+\infty$ for a.e.
$x\in\widehat{\Omega}$ as $n\to+\infty$. On account of hypothesis $(f_{2})$,
for a.e. $x\in\widehat{\Omega}$ we have

(4.11)
$\displaystyle\frac{F(x,u_{n}(x))}{\|u_{n}\|^{p_{2}^{+}}}=\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p_{2}^{+}}}|w_{n}(x)|^{p_{2}^{+}}\to+\infty\
\text{~{}as~{}}n\to+\infty.$

Then by Fatou’s lemma, we obtain

(4.12)
$\int_{\widehat{\Omega}}\frac{F(x,u_{n}(x))}{\|u_{n}\|^{p_{2}^{+}}}dx\to+\infty\text{~{}~{}as~{}}{n\to+\infty}.$

Hypotheses $(f_{1})$-$(f_{2})$ imply there exists $K>0$ such that

(4.13) $\frac{F(x,t)}{|t|^{p_{2}^{+}}}\geq 1\quad\text{for a.e.}\ x\in\Omega,\
\text{all}\ |t|>K.$

By $(f_{1}),$ there exists a positive constant $\widehat{C}>0$ such that

(4.14) $\displaystyle|F(x,t)|\leq\widehat{C},\text{ for
all~{}}(x,t)\in\overline{\Omega}\times[-K,K].$

Now from (4.13) and (4.14), we get

(4.15) $\displaystyle F(x,t)>C_{0}\text{ ~{}for
all~{}}(x,t)\in\overline{\Omega}\times\mathbb{R},$

where $C_{0}\in\mathbb{R}$ is a constant. The above relation implies

$\displaystyle\frac{F(x,u_{n}(x))-C_{0}}{\|u_{n}\|^{p_{2}^{+}}}\geq
0\text{~{}~{} for all~{}}x\in\overline{\Omega},\text{ for
all~{}}n\in\mathbb{N}.$

that is,

(4.16)
$\displaystyle\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p_{2}^{+}}}|w_{n}(x)|^{p_{2}^{+}}-\frac{C_{0}}{\|u_{n}\|^{p_{2}^{+}}}\geq
0\text{~{}~{} for all~{}}x\in\overline{\Omega},\text{ for
all~{}}n\in\mathbb{N}.$

By (4.3), (4.9), (4.12), (4.16) and using the fact $\|w_{n}\|=1,$ Lemma 3.1
and Fatou’s lemma, we have

$\displaystyle+\infty$
$\displaystyle=\left[\int_{\widehat{\Omega}}\liminf_{n\to+\infty}\frac{F(x,u_{n}(x))|w_{n}(x)|^{p_{2}^{+}}}{|u_{n}(x)|^{p_{2}^{+}}}dx-\int_{\widehat{\Omega}}\limsup_{n\to+\infty}\frac{C_{0}}{\|u_{n}\|^{p_{2}^{+}}}dx\right]$
$\displaystyle=\int_{\widehat{\Omega}}\liminf_{n\to+\infty}\left[\frac{F(x,u_{n}(x))|w_{n}(x)|^{p_{2}^{+}}}{|u_{n}(x)|^{p_{2}^{+}}}-\frac{C_{0}}{\|u_{n}\|^{p_{2}^{+}}}\right]dx$
$\displaystyle\leq\liminf_{n\to+\infty}\int_{\widehat{\Omega}}\left[\frac{F(x,u_{n}(x))|w_{n}(x)|^{p_{2}^{+}}}{|u_{n}(x)|^{p_{2}^{+}}}-\frac{C_{0}}{\|u_{n}\|^{p_{2}^{+}}}\right]dx$
$\displaystyle\leq\liminf_{n\to+\infty}\int_{{\Omega}}\left[\frac{F(x,u_{n}(x))|w_{n}(x)|^{p_{2}^{+}}}{|u_{n}(x)|^{p_{2}^{+}}}-\frac{C_{0}}{\|u_{n}\|^{p_{2}^{+}}}\right]dx$
$\displaystyle=\left[\liminf_{n\to+\infty}\int_{{\Omega}}\frac{F(x,u_{n}(x))|w_{n}(x)|^{p_{2}^{+}}}{|u_{n}(x)|^{p_{2}^{+}}}dx-\limsup_{n\to+\infty}\int_{{\Omega}}\frac{C_{0}}{\|u_{n}\|^{p_{2}^{+}}}dx\right]$
$\displaystyle=\liminf_{n\to+\infty}\int_{{\Omega}}\frac{F(x,u_{n}(x))}{\|u_{n}\|^{p_{2}^{+}}}dx$
$\displaystyle=\liminf_{n\to+\infty}\bigg{[}\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{1}{\|u_{n}\|^{p_{2}^{+}-p_{1}(x,y)}}\frac{|w_{n}(x)-w_{n}(y)|^{p_{1}(x,y)}}{p_{1}(x,y)|x-y|^{N+s(x,y)p_{1}(x,y)}}dxdy$
$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{1}{\|u_{n}\|^{p_{2}^{+}-p_{2}(x,y)}}\frac{|w_{n}(x)-w_{n}(y)|^{p_{2}(x,y)}}{p_{2}(x,y)|x-y|^{N+s(x,y)p_{2}(x,y)}}dxdy$
$\displaystyle\quad+\int_{\Omega}\frac{1}{\|u_{n}\|^{p_{2}^{+}-\overline{p}_{1}(x)}}\frac{|w_{n}|^{\overline{p}_{1}(x)}}{\overline{p}_{1}(x)}dx+\int_{\Omega}\frac{1}{\|u_{n}\|^{p_{2}^{+}-\overline{p}_{2}(x)}}\frac{|w_{n}|^{\overline{p}_{2}(x)}}{\overline{p}_{2}(x)}dx$
$\displaystyle\quad+\int_{\mathcal{C}\Omega}\frac{\beta(x)|w_{n}|^{\overline{p}_{1}(x)}}{\|u_{n}\|^{p_{2}^{+}-\overline{p}_{1}(x)}\overline{p}_{1}(x)}\,dx+\int_{\mathcal{C}\Omega}\frac{\beta(x)|w_{n}|^{\overline{p}_{2}(x)}}{\|u_{n}\|^{p^{+}-\overline{p}_{2}(x)}\overline{p}_{2}(x)}dx-\frac{\mathcal{I}(u_{n})}{\|u_{n}\|^{p^{+}}}\bigg{]}$
(4.17) $\displaystyle\leq\liminf_{n\to+\infty}\rho(w_{n})=1.$

Thus we arrive at a contradiction. Hence, $w=0$. Let $\mu\geq 1$ and set
$\kappa:=(2\mu)^{\frac{1}{p_{2}^{-}}}\geq 1$ for all $n\in\mathbb{N}$.
Evidently, from (4.10) we have

$w_{n}\to 0\quad\text{~{} strongly in}\quad
L^{\gamma(\cdot)}(\Omega),~{}1<\gamma(x)<{p_{2}}_{s}^{*}(x)$

which combining with $(f_{1})$-$(f_{3})$ and Lebesgue dominated convergence
theorem yields that

(4.18) $\displaystyle\int_{\Omega}F(x,\kappa w_{n})\,dx\to 0\ \text{as}\
n\to+\infty.$

We can find $t_{n}\in[0,1]$ such that

(4.19) $\mathcal{I}(t_{n}u_{n})=\max_{0\leq t\leq 1}\mathcal{I}(tu_{n}).$

Because of (4.9), for sufficiently large $n\in\mathbb{N},$ we have

(4.20) $0<\frac{(2\mu)^{\frac{1}{p_{2}^{-}}}}{\|u_{n}\|}\leq 1.$

Using (4.18), (4.19) and (4.20) and recalling that $\|w_{n}\|=1,$ for
sufficiently large $n\in\mathbb{N},$ it follows that

$\displaystyle\mathcal{I}(t_{n}u_{n})\geq\mathcal{I}\left(\kappa\frac{u_{n}}{\|u_{n}\|}\right)$
$\displaystyle=\mathcal{I}(\kappa w_{n})$
$\displaystyle\geq(\kappa)^{p_{2}^{-}}\frac{1}{2}\rho(w_{n})-\int_{\Omega}F(x,\kappa
w_{n})dx$ $\displaystyle=2\mu.\frac{1}{2}\rho(w_{n})-\int_{\Omega}F(x,\kappa
w_{n})dx$ $\displaystyle=\mu+o_{n}(1).$

Since $\mu>0$ is arbitrary, we have

(4.21) $\mathcal{I}(t_{n}u_{n})\to+\infty\quad\text{as}\quad n\to+\infty.$

From the assumption$f(x,0)=0$ and (4.3) we know that

(4.22) $\mathcal{I}(0)=0\ \text{and}\ \mathcal{I}(u_{n})\leq C\ \text{for
all}\ n\in\mathbb{N}.$

By (4.21) and (4.22), we can infer that, for $n\in\mathbb{N}$ large,

(4.23) $t_{n}\in(0,1).$

From (4.19) and (4.23), we can see that for all $n\in\mathbb{N}$ sufficiently
large,

(4.24) $\displaystyle
0=t_{n}\frac{d}{dt}\mathcal{I}(tu_{n})|_{t=t_{n}}=\langle\mathcal{I}^{{}^{\prime}}(t_{n}u_{n}),t_{n}u_{n}\rangle,$

so,

$\displaystyle\frac{1}{2}\delta^{p_{1}}_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}(t_{n}u_{n})+\rho^{p_{1}}_{\Omega}(t_{n}u_{n})+\int_{\mathcal{C}\Omega}\beta(x)|t_{n}u_{n}|^{\overline{p}_{1}(x)}\,dx$
(4.25)
$\displaystyle+\frac{1}{2}\delta^{p_{2}}_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}(t_{n}u_{n})+\rho^{p_{2}}_{\Omega}(t_{n}u_{n})+\int_{\mathcal{C}\Omega}\beta(x)|t_{n}u_{n}|^{\overline{p}_{2}(x)}\,dx-\int_{\Omega}f(x,t_{n}u_{n})t_{n}u_{n}\,dx=0.$

From hypothesis $(f_{4})$, we obtain for all $n\in\mathbb{N},$

$\mathcal{F}(x,t_{n}u_{n})\leq\mathcal{F}(x,u_{n})+b(x)\ \text{for a.e}\
x\in\Omega,$

that is,

(4.26)
$f(x,t_{n}u_{n})(t_{n}u_{n})\leq\mathcal{F}(x,u_{n})+b(x)+p_{2}^{+}F(x,t_{n}u_{n})\
\text{for a.e}\ x\in\Omega.$

Combining (4.3) and (4.26), we deduce

$\displaystyle\frac{1}{2}\delta^{p_{1}}_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}(t_{n}u_{n})+\rho^{p_{1}}_{\Omega}(t_{n}u_{n})+\int_{\mathcal{C}\Omega}\beta(x)|t_{n}u_{n}|^{\overline{p}_{1}(x)}\,dx$
$\displaystyle+\frac{1}{2}\delta^{p_{2}}_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}(t_{n}u_{n})+\rho^{p_{2}}_{\Omega}(t_{n}u_{n})+\int_{\mathcal{C}\Omega}\beta(x)|t_{n}u_{n}|^{\overline{p}_{2}(x)}\,dx-
p_{2}^{+}\int_{\Omega}F(x,t_{n}u_{n})dx$
$\displaystyle\leq\int_{\Omega}\mathcal{F}(x,u_{n})dx+\|b\|_{L^{1}(\Omega)}\
\text{for all}\ n\in\mathbb{N},$

and hence by (4.8), we get

(4.27) $p_{2}^{+}\mathcal{I}(t_{n}u_{n})\leq C\ \text{for all}\
n\in\mathbb{N}.$

We compare (4.21) and (4.27) and arrive at a contradiction. Thus the claim
follows.

On account of this claim, we may assume that

(4.28) $u_{n}\rightharpoonup u\ \text{~{} weakly in}\ X\quad\text{and}\quad
u_{n}\to u\ \text{~{} strongly in}\ L^{\gamma(\cdot)}(\Omega),\
1<\gamma(x)<{p_{2}}_{s}^{*}(x).$

We show in what follows that

$u_{n}\to u\ \text{in}\ X.$

Using (4.28), we have

(4.29) $\displaystyle
o_{n}(1)=\langle\mathcal{I}^{{}^{\prime}}(u_{n}),u_{n}-u\rangle\geq\frac{1}{2}\langle\rho^{\prime}(u_{n}),u_{n}-u\rangle-\int_{\Omega}f(x,u_{n})(u_{n}-u)dx.$

Now by $(f_{1}),$ Hölder inequality, (4.28), boundedness of $(u_{n})_{n}$ and
Lemma 2.3, we obtain

$\displaystyle\left|\int_{\Omega}f(x,u_{n})(u_{n}-u)dx\right|$
$\displaystyle\leq\|a\|_{L^{\infty}(\Omega)}\left[\int_{\Omega}|u_{n}-u|.1\,dx+\int_{\Omega}|v_{n}|^{r(x)-1}|v_{n}-v_{0}|dx\right]$
$\displaystyle\leq\|a\|_{L^{\infty}(\Omega)}\left[\|u_{n}-u\|_{L^{r(\cdot)}(\Omega)}\left(1+|\Omega|\right)^{r^{\prime+}}+\|v_{n}-v_{0}\|_{L^{r(\cdot)}(\Omega)}\||u_{n}|^{r(\cdot)-1}\|_{L^{r^{\prime}(\cdot)}(\Omega)}\right]$
$\displaystyle\leq
C\left[\|u_{n}-u\|_{L^{r(\cdot)}(\Omega)}+\|u_{n}-u\|_{L^{r(\cdot)}(\Omega)}\left(\|u_{n}\|_{L^{r(\cdot)}(\Omega)}^{r^{+}-1}+\|u_{n}\|_{L^{r(\cdot)}(\Omega)}^{r^{-}-1}\right)\right]$
$\displaystyle\quad\to 0\text{\;\; as\;}n\to+\infty.$

Hence, combining (4.29) and (4.3) and using the $(S_{+})$ property of
$\rho^{\prime}$ (see Lemma 3.5), we have $u_{n}\to u$ strongly in $X$ as
$n\to+\infty.$, which shows that the $(C)_{c}$-condition is satisfied. The
proof is now complete. ∎

##  5\. Proof of Theorem 1.2

To prove the Theorem 1.2 we need the Fountain theorem of Bartsch [5, Theorem
2.5]; (see also [46, Theorem 3.6]). We recall next lemma from [20].

######  Lemma 5.1.

Let $E$ be a reflexive and separable Banach space. Then there are
$\\{e_{n}\\}\subset E$ and $\\{g_{n}^{*}\\}\subset E^{*}$ such that

$E=\overline{span\\{e_{n}:n=1,2,3..\\}},~{}~{}E^{*}=\overline{span\\{g_{n}^{*}:n=1,2,3..\\}},$

and

$\langle
g_{i}^{*},e_{j}\rangle=\left\\{\begin{array}[]{rl}&1\text{~{}~{}~{}~{}if~{}~{}~{}}i=j\\\
&0\text{~{}~{}~{}~{}if~{}~{}~{}}i\not=j.\end{array}\right.$

Let us denote

(5.1) $\displaystyle
E_{n}=span\\{e_{n}\\},~{}~{}~{}X_{k}=\bigoplus_{n=1}^{k}E_{n}\text{
~{}~{}~{}and }Y_{k}=\overline{\bigoplus_{n=k}^{\infty}E_{n}}.$

Now we recall the following Fountain theorem from [2]:

######  Theorem 5.2 (Fountain theorem).

Assume that $\Phi\in C^{1}(E,\mathbb{R})$ satisfies the Cerami condition
$(C)_{c}$ for all $c\in\mathbb{R}$ and $\Phi(-u)=\Phi(u).$ If for each
sufficiently large $k\in\mathbb{N},$ there exists $\varrho_{k}>\delta_{k}>0$
such that

  * ${(\mathcal{A}_{1})}$

$b_{k}:=\inf\\{\Phi(u):u\in Y_{k},~{}\|u\|_{E}=\delta_{k}\\}\to+\infty,$ as
$k\to+\infty,$

  * $\rm(\mathcal{A}_{2})$

$a_{k}:=\max\\{\Phi(u):u\in X_{k},~{}\|u\|_{E}=\varrho_{k}\\}\leq 0.$

Then $\Phi$ has a sequence of critical points $(u_{k})_{k}$ such that
$\Phi(u_{k})\to+\infty.$

###### Proof of Theorem 1.2:.

Define $X_{k}$ and $Y_{k}$ as in (5.1) for the reflexive, separable Banach
space $X$. Now $\mathcal{I}$ is even and satisfies Cerami condition $(C)_{c}$
for all $c\in\mathbb{R}$ (see Lemma 4.4). So now we will show that the
conditions $(\mathcal{A}_{1})$-$(\mathcal{A}_{2})$ hold for our problem.

  * $(\mathcal{A}_{1}):$

For large $k\in\mathbb{N},$ set

(5.2) $\displaystyle\alpha_{k}=\sup_{u\in
Y_{k},~{}\|u\|=1}\|u\|_{L^{\gamma(\cdot)}(\Omega)},$

where $\gamma\in C_{+}(\overline{\Omega})$ such that for all
$x\in\overline{\Omega},$ $1<\gamma(x)<p_{s}^{*}(x).$ So,

(5.3) $\displaystyle\lim_{k\to+\infty}\alpha_{k}=0.$

Supposing to the contrary, there exist $\epsilon^{\prime}>0,k_{0}\geq 0$ and a
sequence $(u_{k})_{k}$ in $Y_{k}$ such that

$\|u_{k}\|=1\text{
and}~{}~{}\|u_{k}\|_{L^{\gamma(\cdot)}(\Omega)}\geq\epsilon^{\prime}$

for all $k\geq k_{0}.$ Since $(u_{k})_{k}$ is bounded in $X,$ there exists
$u_{0}\in E$ such that up to a subsequence, still denoted by $(u_{k})_{k},$ we
have $u_{k}\rightharpoonup u_{0}$ weakly in $E$ as $k\to+\infty$ and

$\langle g_{j}^{*},u_{0}\rangle=\lim_{k\to+\infty}\langle
g_{j}^{*},u_{k}\rangle=0$

for $j=1,2,3,\cdots.$ Thus we have $u=0.$ In addition, using Theorem 3.4 we
obtain

$\epsilon^{\prime}\leq\lim_{k\to+\infty}\|u_{k}\|_{L^{\gamma(\cdot)}(\Omega)}=\|u_{0}\|_{L^{\gamma(\cdot)}(\Omega)}=0,$

a contradiction. Hence, (5.3) holds true. Let $u\in Y_{k}$ with $\|u\|>1.$
Note that (5.3) implies $\alpha_{k}<1$ for large $k\in\mathbb{N}.$ Thus using
Lemma 2.3, Lemma 3.3 and (5.2) and (4.1) with $\epsilon=1$ for
$k\in\mathbb{N}$ large enough, we get

$\displaystyle\mathcal{I}(u)$
$\displaystyle\geq\frac{1}{2}\rho(u)-\left[\int_{\Omega}|u|^{p_{2}^{+}}\,dx-C(1)\int_{\Omega}|u|^{r(x)}\,dx\right]$
$\displaystyle\geq\frac{1}{2}\|u\|^{p_{1}^{-}}-\|u\|^{p_{2}^{+}}_{L^{p_{2}^{+}}(\Omega)}-C(1)\left\\{\|u\|^{r^{-}}_{L^{r(\cdot)}(\Omega)}+\|u\|^{r^{+}}_{L^{r(\cdot)}(\Omega)}\right\\}$
$\displaystyle\geq\frac{1}{2}\|u\|^{p_{1}^{-}}-\alpha_{k}^{p_{2}^{+}}C_{1}\|u\|^{p_{2}^{+}}-C_{2}\\{\alpha_{k}^{r^{-}}\|u\|^{r^{-}}+\alpha_{k}^{r^{+}}\|u\|^{r^{+}}\\}$
(5.4)
$\displaystyle\geq\frac{1}{2}\|u\|^{p_{1}^{-}}-C\alpha_{k}\|u\|^{r^{+}},$

where $C,C_{1},C_{2}$ are some positive constants.

Define the function $\mathcal{G}:\mathbb{R}\to\mathbb{R},$

$\mathcal{G}(t)=\frac{1}{2}t^{p_{1}^{-}}-C\alpha_{k}t^{r^{+}}.$

Then it can be derived by a simple computation that $G$ attains its maximum at

$\delta_{k}={\left(\frac{p_{1}^{-}}{2r^{+}{C}\alpha_{k}}\right)^{1/(r^{+}-p_{1}^{-})}}$

and the maximum value of $\mathcal{G}$ is

$\displaystyle\mathcal{G}(\delta_{k})$
$\displaystyle=\frac{1}{2}\left(\frac{p_{1}^{-}}{2r^{+}{C}\alpha_{k}}\right)^{{p_{1}^{-}}/(r^{+}-p_{1}^{-})}-{{C}}\alpha_{k}\left(\frac{p_{1}^{-}}{2r^{+}{C}\alpha_{k}}\right)^{{r^{+}}/(r^{+}-p_{1}^{-})}$
$\displaystyle=\left(\frac{1}{2}\right)^{r^{+}/(r^{+}-p_{1}^{-})}\left(\frac{1}{{{C}}\alpha_{k}}\right)^{{p_{1}^{-}}/(r^{+}-p_{1}^{-})}\left(\frac{p_{1}^{-}}{r^{+}}\right)^{\theta
p_{1}^{-}/(r^{+}-p_{1}^{-})}\bigg{(}1-\frac{p_{1}^{-}}{r^{+}}\bigg{)}.$

Since $p_{1}^{-}<r^{+}$ and $\alpha_{k}\to 0$ as $k\to+\infty,$ we obtain

(5.5) $\mathcal{G}(\delta_{k})\to+\infty\ \text{as}\ k\to+\infty.$

Again, (5.3) infers $\delta_{k}\to+\infty$ as $k\to+\infty.$ Thus for $u\in
Y_{k}$ with $\|u\|=\delta_{k},$ taking into account ($(\mathcal{A}_{1}):$) and
(5.5), it follows that as $k\to+\infty$

$b_{k}=\inf_{u\in Y_{k},\|u\|=\delta_{k}}\mathcal{I}(u)\to+\infty.$

  * $(\mathcal{A}_{2}):$

Let us assume that the assertion $(\mathcal{A}_{2})$ does not hold for some
$k.$ So there exists a sequence $(u_{n})_{n}\subset X_{k}$ such that

(5.6) $\displaystyle\|u_{n}\|\to+\infty\ \text{and}\ \mathcal{I}(u_{n})\geq
0.$

Let us take $w_{n}:=\frac{u_{n}}{\|u_{n}\|},$ then $w_{n}\in X$ and
$\|w_{n}\|=1.$ Since $X_{k}$ is of finite dimension, there exists $w\in
X_{k}\setminus\\{0\\}$ such that up to a subsequence, still denoted by
$(w_{n})_{n},$ $w_{n}\to w$ strongly and $w_{n}(x)\to w(x)$ a.e. $x\in\Omega$
as $n\to+\infty.$ If $w(x)\not=0$ then $|u_{n}(x)|\to+\infty$ as
$n\to+\infty.$ Similar to (4.11), it follows that for each $x\in\Omega$,

(5.7)
$\displaystyle\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p_{2}^{+}}}|w_{n}(x)|^{p_{2}^{+}}\to+\infty.$

Hence, using (5.6) and applying Fatou’s lemma, we have

(5.8)
$\frac{1}{\|u_{n}\|^{p_{2}^{+}}}\int_{\Omega}F(x,u_{n})dx=\int_{\Omega}\frac{F(x,u_{n})}{|u_{n}(x)|^{p_{2}^{+}}}|w_{n}(x)|^{p_{2}^{+}}dx\to+\infty\
\text{as}\ $

Since $\|u_{n}\|>1$ for large $n\in\mathbb{N}$, from Proposition 3.3 and
(5.8), we obtain as $n\to+\infty$

$\displaystyle\mathcal{I}(u_{n})$
$\displaystyle\leq\rho(u_{n})-\int_{\Omega}F(x,u_{n})\,dx$
$\displaystyle\leq\|u_{n}\|^{p_{2}^{+}}-\int_{\Omega}F(x,u_{n})\,dx$
$\displaystyle=\bigg{(}1-\frac{1}{\|u_{n}\|^{p_{2}^{+}}}\int_{\Omega}F(x,u_{n})\;dx\bigg{)}\|u_{n}\|^{p_{2}^{+}}\to-\infty,$

a contradiction to (5.6). Therefore, for sufficiently large $k\in\mathbb{N},$
we can get $\varrho_{k}>\delta_{k}>0$ such that $(\mathcal{A}_{2})$ holds for
$u\in X_{k}$ with $\|u\|=\varrho_{k}$.

∎

##  6\. Proof of Theorem 1.3

For proving Theorem 1.3, we first recall the Dual fountain theorem due to
Bartsch and Willem (see [46, Theorem 3.18]). Considering Lemma 5.1 and using
the reflexivity and separability of the Banach space $X$ we can define $X_{k}$
and $Y_{k}$ appropriately.

######  Definition 6.1.

For $c\in\mathbb{R},$ we say that $\mathcal{I}$ satisfies the $(C)_{c}^{*}$
condition (with respect to $Y_{k}$) if any sequence $(u_{k})_{k}$ in $X$ with
$u_{k}\in Y_{k}$ such that

$\mathcal{I}(u_{k})\to
c\text{~{}~{}and~{}~{}}\|\mathcal{I}^{\prime}_{|_{Y_{k}}}(u_{k})\|_{E^{*}}(1+\|u_{k}\|)\to
0,\text{~{}~{}as~{}~{}~{}}k\to+\infty$

contains a subsequence converging to a critical point of $\mathcal{I},$ where
$X^{*}$ is the dual of $X.$

######  Theorem 6.2 (Dual fountain Theorem).

Let $\Phi\in C^{1}(E,\mathbb{R})$ satisfy $\Phi(-u)=\Phi(u).$ If for each
$k\geq k_{0}$ there exist $\varrho_{k}>\delta_{k}>0$ such that

  * $(\mathcal{B}_{1})$

$a_{k}=\inf\\{\Phi(u):u\in Z_{k},\,\,\|u\|_{E}=\varrho_{k}\\}\geq 0;$

  * $(\mathcal{B}_{2})$

$b_{k}=\sup\\{\Phi(u):u\in Y_{k},\,\,\|u\|_{E}=\delta_{k}\\}<0;$

  * $(\mathcal{B}_{3})$

$d_{k}=\inf\\{\Phi(u):u\in Z_{k},\,\,\|u\|_{E}\leq\varrho_{k}\\}\to 0$ as
$k\to+\infty;$

  * $(\mathcal{B}_{4})$

$\Phi$ satisfies the $(C)_{c}^{*}$ condition for every $c\in[d_{k_{0}},0[.$

Then $\Phi$ has a sequence of negative critical values converging to $0$.

######  Remark 6.3.

Note that, in [46], assuming that the energy functional associated to the
problem satisfies $(PS)_{c}^{*}$ condition the Dual fountain theorem is
obtained using Deformation theorem which is still valid under Cerami
condition. Therefore, like many critical point theorems the Dual fountain
theorem holds under $(C)_{c}^{*}$ condition.

Next lemma is due to [26, Lemma 3.2]

######  Lemma 6.4.

Suppose that the hypotheses in Theorem 1.3 hold, then $\mathcal{I}$ satisfies
the $(C)_{c}^{*}$ condition.

###### Proof of Theorem 1.3.

For the reflexive, separable Banach space $X,$ define $X_{k}$ and $Y_{k}$ as
in (5.1). From the assumptions we have that $\mathcal{I}$ is even and by Lemma
6.4 we get that $\mathcal{I}$ satisfies Cerami condition $(C)_{c}^{*}$ for all
$c\in\mathbb{R}.$ Thus for proving Theorem 1.3 we are only left with verifying
the conditions $(\mathcal{B}_{1})$-$(\mathcal{B}_{3}).$
$(\mathcal{B}_{1})$: For all $u\in Y_{k}$ with $\|u\|<1,$ arguing similarly as
we did for obtaining ($(\mathcal{A}_{1}):$), we can derive

$\displaystyle\mathcal{I}(u)$
$\displaystyle\geq\frac{1}{2}\left[\rho_{p_{1}}(u)+\rho_{p_{2}}(u)\right]-\int_{\Omega}F(x,u)\,dx$
$\displaystyle\geq\frac{1}{2}\rho(u)-\left[\int_{\Omega}|u|^{p_{2}^{+}}\,dx-C(1)\int_{\Omega}|u|^{r(x)}\,dx\right]$
$\displaystyle\geq\frac{1}{2}\|u\|^{p_{2}^{+}}-\|u\|^{p_{2}^{+}}_{L^{p_{2}^{+}}(\Omega)}-C(1)\left\\{\|u\|^{r^{-}}_{L^{r(\cdot)}(\Omega)}+\|u\|^{r^{+}}_{L^{r(\cdot)}(\Omega)}\right\\}$
$\displaystyle\geq\frac{1}{2}\|u\|^{p_{2}^{+}}-\alpha_{k}^{p_{2}^{+}}C_{1}\|u\|^{p_{2}^{+}}-C_{2}\\{\alpha_{k}^{r^{-}}\|u\|^{r^{-}}+\alpha_{k}^{r^{+}}\|u\|^{r^{+}}\\}$
(6.1) $\displaystyle\geq\frac{1}{2}\|u\|^{p_{2}^{+}}-C_{4}\alpha_{k}\|u\|,$

Let us choose
$\varrho_{k}=\left(C_{4}\alpha_{k}/{2}\right)^{1/{(p_{2}^{+}-1)}}.$ Since
$p_{2}^{+}>1,$ (5.2) infers that

(6.2) $\varrho_{k}\to 0\text{~{}~{}as~{}}k\to+\infty.$

Thus for $u\in Y_{k}$ with $\|u\|=\varrho_{k}$ and for sufficiently large
$k\in\mathbb{N},$ from (6) we have $\mathcal{I}(u)\geq 0.$
$(\mathcal{B}_{2})$: Suppose assertion $(\mathcal{B}_{2})$ does not hold true
for some given $k\in\mathbb{N}.$ Then there exists a sequence $(v_{n})_{n}$ in
$X_{k}$ such that

(6.3) $\displaystyle\|v_{n}\|\to+\infty,~{}~{}~{}~{}~{}\mathcal{I}(v_{n})\geq
0.$

Now arguing in a similar way as in the proof of assertion $(\mathcal{A}_{2})$
of Theorem 5.2, we obtain (5.7) and (5.8) which combining with Lemma 3.3 imply
that

$\displaystyle\mathcal{I}(v_{n})$
$\displaystyle\leq\rho(v_{n})-\int_{\Omega}F(x,v_{n})\,dx$
$\displaystyle\leq{\|v_{n}\|}^{p_{2}^{+}}-\int_{\Omega}F(x,v_{n})\,dx$
$\displaystyle=\bigg{(}1-\frac{1}{\|v_{n}\|^{p_{2}^{+}}}\int_{\Omega}F(x,v_{n})\;dx\bigg{)}\|v_{n}\|^{p_{2}^{+}}\to-\infty\text{\;\;as}$

a contradiction to (6.3). So, there exists $k_{0}\in\mathbb{N}$ such that for
all $k\geq k_{0}$ we have $1>\varrho_{k}>\delta_{k}>0$ such that for $u\in
X_{k}$ with $\|u\|=\delta_{k}$ the condition $(\mathcal{B}_{2})$ holds true.
$(\mathcal{B}_{3})$: Since $X_{k}\cap Y_{k}\not=\emptyset,$ we get that
$d_{k}\leq b_{k}<0.$ Now for $u\in Y_{k}$ with $\|u\|\leq\varrho_{k}$ by (6),
we get

$\displaystyle\mathcal{I}(u)\geq-C_{4}\alpha_{k}\|u\|\geq-
C_{4}\alpha_{k}\varrho_{k}.$

Therefore, combining (5.2) and (6.2), we obtain

$d_{k}\geq-C_{4}\alpha_{k}\varrho_{k}\to 0\text{~{}~{} as~{}}k\to+\infty.$

Since $d_{k}<0,$ it follows that $\lim_{k\to+\infty}d_{k}=0.$ Thus the proof
of the theorem is complete. ∎

##  7\. Proof of Theorem 1.4

First we state the following $\mathbb{Z}_{2}$-symmetric version of mountain
pass theorem due to [38, Theorem 9.12]. Here again we want to mention that in
[38] this theorem is proved using $(PS)$-condition, which can also be proved
using $C$-condition.

######  Theorem 7.1.

(Symmetric Mountain pass Theorem): Let $E$ be a real infinite dimensional
Banach space and $\Phi\in C^{1}(E,\mathbb{R})$ be an even functional
satisfying the $(C)_{c}$ condition. Also let $\Phi$ satisfy the following:

  * $(\mathcal{D}_{1})$

$\Phi(0)=0$ and there exist two constants $\nu,\mu>0$ such that
$\Phi(u)\geq\mu$ for all $u\in E$ with $\|u\|_{E}=\nu.$

  * $(\mathcal{D}_{2})$

for all finite dimensional subspaces $\widehat{E}\subset E$ there exists
$\overline{R}=\overline{R}(\widehat{E})>0$ such that $\Phi(u)\leq 0$ for all
$u\in\widehat{E}\setminus B_{\overline{R}}(\widehat{E}),$ where
$B_{\overline{R}}(\widehat{E})=\\{u\in\widehat{E}:\|u\|_{E}\leq\overline{R}\\}.$

Then $\Phi$ poses an unbounded sequence of critical values characterized by a
minimax argument.

###### Proof of Theorem 1.4 .

From the hypotheses of the theorem it follows that $\mathcal{I}$ is even and
we have $\mathcal{I}(0)=0.$ Now we will prove that $\mathcal{I}$ satisfies the
assertions in Theorem 7.1.

  * $(\mathcal{D}_{1}):$

It follows from Lemma 4.3$(i)$.

  * $(\mathcal{D}_{2}):$

To show this, first claim that for any finite dimensional subspace $Y$ of $X$
there exists $\overline{R}_{0}=\overline{R}_{0}(Y)$ such that
$\mathcal{I}(u)<0$ for all $u\in E\setminus B_{\overline{R}_{0}}(Y),$ where
$B_{\overline{R}_{0}}(Y)=\\{u\in X:\|u\|\leq\overline{R}_{0}\\}.$ Fix $u\in
X,\;\|u\|=1.$ For $t>1$ using (4.2) and Lemma 3.3, we get

$\displaystyle\mathcal{I}(tu)$
$\displaystyle\leq\rho(tu)-\int_{\Omega}F(x,u)dx$ $\displaystyle\leq
t^{p_{2}^{+}}\rho(u)-t^{p_{2}^{+}}M\int_{\Omega}|u|^{p_{2}^{+}}\,dx+|\Omega|C_{M}$
(7.1)
$\displaystyle=t^{p_{2}^{+}}\left[1-M\|u\|_{L^{p_{2}^{+}}(\Omega)}^{p_{2}^{+}}\,\right]+|\Omega|C_{M}.$

Since $Y$ is finite dimensional, all norms are equivalent on $Y$, which infers
that there exists some constant $C(Y)>0$ such that
$C(Y)\|u\|\leq\|u\|_{L^{p_{2}^{+}}(\Omega)}.$ Therefore, from
($(\mathcal{D}_{2}):$), we obtain

$\displaystyle\mathcal{I}(tu)$ $\displaystyle\leq
t^{p_{2}^{+}}\left[1-M(C(Y))^{p_{2}^{+}}\|u\|^{p_{2}^{+}}\,\right]+|\Omega|C_{M}$
$\displaystyle=t^{p_{2}^{+}}\left[1-M(C(Y))^{p_{2}^{+}}\,\right]+|\Omega|C_{M}.$

Now by choosing $M$ sufficiently large such that
$M>\frac{1}{(C(Y))^{p_{2}^{+}}},$ from the last relation we yields that

$\mathcal{I}(u)\to-\infty\text{\;\;\; as\;\;}t\to+\infty.$

Hence, there exists $\overline{R}_{0}>0$ large enough such that
$\mathcal{I}(u)<0$ for all $u\in X$ with $\|u\|=\overline{R}$ and
$\overline{R}\geq\overline{R}_{0}$. Therefore, $\mathcal{I}$ verifies
$(\mathcal{D}_{2})$.

∎

##  8\. Proof of Theorem 1.5

First, we recall a new variant of Clark’s theorem (see [31, Theorem 1.1]).

######  Theorem 8.1.

Let $E$ be a Banach space, $\Phi\in C^{1}(E,\mathbb{R})$. Let $\Phi$ be even
and $\Phi(0)=0$. Also assume $\Phi$ satisfies the $(PS)$-condition and bounded
from below. If for any $k\in\mathbb{N},$ there exists a $k$-dimensional
subspace $E^{k}$ of $E$ and $\beta_{k}>0$ such that
$\displaystyle\sup_{E^{k}\cap B_{\beta_{k}}}\Phi(u)<0,$ where
$B_{\beta_{k}}=\\{u\in E:\|u\|_{E}=\beta_{k}\\}$, then at least one of the
following conclusions holds:

  * $(\mathcal{M}_{1})$

There exists a sequence of critical points $(u_{k})_{k}$ satisfying
$\Phi(u_{k})<0$ for all $k$ and $\|u_{k}\|_{E}\to 0$ as $k\to+\infty.$

  * $(\mathcal{M}_{2})$

There exists $l>0$ such that for any $0<b<l$ there exists a critical point $u$
such that $\|u\|_{E}=b$ and $\Phi(u)=0.$

The corresponding energy functional is given as

$\displaystyle\mathcal{I}(u)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{1}(x,y)}}{p_{1}(x,y)|x-y|^{N+s(x,y)p_{1}(x,y)}}dxdy+\int_{\Omega}\frac{1}{\overline{p}_{1}(x)}|u|^{\overline{p}_{1}(x)}dx$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{2}(x,y)}}{p_{2}(x,y)|x-y|^{N+s(x,y)p_{2}(x,y)}}dxdy+\int_{\Omega}\frac{1}{\overline{p}_{2}(x)}|u|^{\overline{p}_{2}(x)}dx$
$\displaystyle+\int_{\mathcal{C}\Omega}\frac{\beta(x)|u|^{\overline{p}_{1}(x)}}{\overline{p}_{1}(x)}\,dx+\int_{\mathcal{C}\Omega}\frac{\beta(x)|u|^{\overline{p}_{2}(x)}}{\overline{p}_{2}(x)}\,dx$
$\displaystyle\quad-\lambda\int_{\Omega}\frac{|u|^{q(x)}}{q(x)}dx-\int_{\Omega}\frac{|u|^{r(x)}}{r(x)}dx.$

Next, we will prove the following lemma:

######  Lemma 8.2.

Suppose the hypotheses in Theorem 1.5 hold. Then $\mathcal{I}$ satisfies
$(PS)_{c}$ for any $c\in\mathbb{R}.$

###### Proof.

Let $(v_{n})_{n}$ be a sequence in $X$ such that

(8.1) $\displaystyle\mathcal{I}(v_{n})\to
c\text{\;\;\;and\;\;}\mathcal{I}^{\prime}(v_{n})\to
0\text{\;in\;}X^{*}\text{\;\; as\;\;}n\to+\infty.$

Therefore,

(8.2) $\displaystyle\langle\mathcal{I}^{\prime}(v_{n}),v_{n}-v_{0}\rangle\to
0\text{\;\;as\;}n\to+\infty.$

Hence, we have that $(v_{n})_{n}$ is bounded in $X.$ If not, then
$v_{n}\to+\infty$ as $n\to+\infty.$ Using (8.1) and (8.2) and $(f_{5}),$ we
deduce

$\displaystyle 1+C+\|v_{n}\|$
$\displaystyle\geq\mathcal{I}(v_{n})-\frac{1}{q^{-}}\langle\mathcal{I}^{\prime}(v_{n}),v_{n}\rangle$
$\displaystyle\geq\frac{1}{2}\left[\rho(v_{n})-\int_{\Omega}F(x,v_{n})dx-\frac{1}{q^{-}}\rho(v_{n})+\frac{1}{q^{-}}\int_{\Omega}f(x,v_{n})v_{n}dx\right]$
$\displaystyle\geq\frac{1}{2}\left[\left(1-\frac{1}{q^{-}}\right)\|v_{n}\|^{p^{-}}+\left(\frac{1}{q^{-}}-\frac{1}{r^{-}}\right)\int_{\Omega}|v_{n}|^{r(x)}dx\right]$
(8.3)
$\displaystyle\geq\frac{1}{2}\left(1-\frac{1}{q^{-}}\right)\|v_{n}\|^{p^{-}},$

which is a contradiction to the fact that $v_{n}\to+\infty$ as $n\to+\infty$.
Now, since $X$ is reflexive, up to a subsequence, still denoted by
$(v_{n})_{n}$, we have $v_{n}\rightharpoonup v_{0}$ weakly as $n\to+\infty.$
Therefore, as $n\to+\infty$ by Theorem 3.4, arguing similar as in (4.3), we
obtain

(8.4) $\displaystyle v_{n}\to v_{0}\text{\;\;strongly
in\;}L^{\gamma}(\Omega),\,1<\gamma(x)<p_{s}^{*}(x)\text{\;\;
and\;\;}v_{n}(x)\to v_{0}(x)\text{\;a.e. in\;}\Omega.$

By $(f_{5}),$ Hölder inequality, (8.4), boundedness of $(v_{n})_{n}$ and Lemma
2.3, arguing in a similar fashion as (4.3), we obtain

(8.5) $\displaystyle\left|\int_{\Omega}f(x,v_{n})(v_{n}-v_{0})dx\right|\to
0\text{\;\; as\;}n\to+\infty.$

Hence, combining (8.2) and (8.5) and using the $(S_{+})$ property of
$\rho^{\prime},$ we have $v_{n}\to v_{0}$ strongly in $X$ as $n\to+\infty.$

∎

###### Proof of Theorem 1.5:.

From the hypotheses we have that $\mathcal{I}$ is even and $\mathcal{I}(0)=0.$
Also Lemma 8.2 ensures that $\mathcal{I}$ satisfies $(PS)$-condition. But note
that, $\mathcal{I}$ is not bounded from below on $X.$ Hence, we will use a
truncation technique. For that choose $h\in C^{\infty}([0,\infty),[0,1])$ such
that

$h(t)=\begin{cases}&1\mbox{\;\;if}\ t\in[0,l_{0}]\\\ &0\mbox{\;\;if}\
t\in[l_{1},\infty),\end{cases}$

where $l_{0}<l_{1}$ and set $\Psi(u):=h(\|u\|).$ Now we define the truncated
functional $\mathcal{J}$ as:

$\displaystyle\mathcal{J}(u)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{1}(x,y)}}{p_{1}(x,y)|x-y|^{N+s(x,y)p_{1}(x,y)}}dxdy+\int_{\Omega}\frac{1}{\overline{p}_{1}(x)}|u|^{\overline{p}_{1}(x)}dx$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{2N}\setminus(\mathcal{C}\Omega)^{2}}\frac{|u(x)-u(y)|^{p_{2}(x,y)}}{p_{2}(x,y)|x-y|^{N+s(x,y)p_{2}(x,y)}}dxdy+\int_{\Omega}\frac{1}{\overline{p_{2}}(x)}|u|^{\overline{q}_{2}(x)}dx$
$\displaystyle+\int_{\mathcal{C}\Omega}\frac{\beta(x)|u|^{\overline{p}_{1}(x)}}{\overline{p}_{1}(x)}\,dx+\int_{\mathcal{C}\Omega}\frac{\beta(x)|u|^{\overline{p}_{2}(x)}}{\overline{p}_{2}(x)}\,dx$
$\displaystyle\quad-\Psi(u)\left(\lambda\int_{\Omega}\frac{|u|^{q(x)}}{q(x)}dx-\int_{\Omega}\frac{|u|^{r(x)}}{r(x)}dx\right).$

Then $\mathcal{J}\in C^{1}(X,\mathbb{R})$ and $\mathcal{J}(0)=0.$ Also
$\mathcal{J}$ is even. Moreover from Lemma 8.2, it can be shown that
$\mathcal{J}$ satisfies $(PS)$-condition. Now we will show $\mathcal{J}$ is
bounded from below. For $\|u\|>1,$ using Lemma 3.3, we get

$\displaystyle\mathcal{J}(u)\geq\frac{1}{2}\rho(u)\geq\frac{1}{2}\|u\|^{p_{1}^{-}}\to+\infty$

as $\|u\|\to+\infty,$ that is $\mathcal{J}(u)$ is coercive and hence bounded
below on $X.$ Next, we claim that $\mathcal{J}$ verifies the assertion
$(\mathcal{M}_{1})$ of Theorem 8.1. For any $k\in\mathbb{N}$ and
$0<R_{k}<l_{0}<1$ let us set

$\mathcal{B}_{R_{k}}=\\{u\in X:\|u\|=R_{k}\\}.$

Also consider the $k$-dimensional subspace $X^{k}$ of $X$. Then for $u\in
X^{k}\cap\mathcal{B}_{R_{k}}$ there are some constants $K_{1},K_{2}>0$ such
that

$\displaystyle\mathcal{J}(u)$ $\displaystyle\leq\rho(u)-\int_{\Omega}F(x,u)dx$
$\displaystyle\leq\|u\|^{p_{1}^{-}}-\frac{\lambda}{q^{+}}\int_{\Omega}|u|^{q(x)}dx-\frac{1}{r^{+}}\int_{\Omega}|u|^{r(x)}dx$
(8.6)
$\displaystyle\leq\|u\|^{p_{1}^{-}}-K_{1}^{q^{+}}\frac{\lambda}{q^{+}}\|u\|^{q^{+}},$

since $X^{k}\cap\mathcal{B}_{R_{k}}$ being of finite dimension all norms on it
are equivalent. Now by letting $R_{k}\to 0$ as $k\to+\infty$ from (8), we get
$\sup_{X^{k}\cap\mathcal{B}_{\mathbb{R}_{k}}}\mathcal{J}(u)<0$ since
$p_{1}^{-}<q^{+}.$ Furthermore, for a given $u\in X$ from (8) it follows that
$\mathcal{J}(tu)<0$ for $t\to 0^{+},$ that is $\mathcal{J}(u)\not=0.$ Thus
$\mathcal{J}$ does not satisfy $(\mathcal{M}_{2}).$ Therefore, by appealing
Theorem 8.1, there exists a sequence of critical points $(v_{k})_{k}$ of
$\mathcal{J}$ in $X$ such that $\|v_{k}\|\to 0$ as $k\to+\infty.$ So, for
$l_{0}>0$ there exists $\hat{k}_{0}\in\mathbb{N}$ such that for all
$k\geq\hat{k}_{0}$ we have $\|u\|<l_{0}$ which infers that
$\mathcal{J}(u_{k})=\mathcal{I}(u_{k})$ for all $k>\hat{k}_{0}.$ Since the
critical values of $\mathcal{I}$ are the solutions to (1.2), the theorem
follows. ∎

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