# Multiple Solutions for Nonlinear Generalized-Kirchhoff type potential
Systems in Unbounded Domains

Nabil Chems Eddine and Anass Ouannasser<EMAIL_ADDRESS><EMAIL_ADDRESS>Center of Mathematical Research and Applications
of Rabat (CeReMAR), Laboratory of Mathematical Analysis and Applications,
Faculty of Sciences, Mohammed V University, P.O. Box 1014, Rabat, Morocco.

###### Abstract.

In this paper, we consider a class of quasilinear stationary Kirchhoff type
potential systems in unbounded domains, which involves a general variable
exponent elliptic operator. Under some suitable conditions on the
nonlinearities, we establish existence of at least three weak solutions for
the problem. The proof of our main result uses variational methods and the
critical theorem of Bonanno and Marano.

###### Key words and phrases:

Multiple solutions, variable exponent spaces, Kirchhoff-type problems,
$p$-Laplacian, $p(x)$-Laplacian, generalized Capillary operator, critical
points theory.

###### 2010 Mathematics Subject Classification:

35J60, 47J30, 58E05

##  1\. Introduction

The aim of this paper is to show the existence of at least three weak
solutions for the following class of nonlocal quasilinear elliptic systems in
$\mathbb{R}^{N}$

$\displaystyle\begin{cases}-M_{i}\Big{(}\mathcal{B}_{i}(u_{i})\Big{)}\textrm{div}\,\Big{(}\mathcal{A}_{1i}(\nabla
u_{i})+w_{i}(x)\mathcal{A}_{2i}(u_{i})\Big{)}=\lambda F_{u_{i}}(x,u)&\text{in
}\mathbb{R}^{N},\\\ \qquad u_{i}\in W_{w_{i}}^{1,p_{i}(x)}(\mathbb{R}^{N})\cap
W_{w_{i}}^{1,\gamma_{i}(x)}(\mathbb{R}^{N});\end{cases}$ (1.1)

for $1\leq i\leq n$ ($n\in\mathbb{N}$), where $N\geq 2$, $\lambda$ is a
positive real parameter and the functions $w_{i}\in
L^{\infty}(\mathbb{R}^{N})$ such that $\displaystyle
w_{i}:=ess\inf_{x\in\mathbb{R}^{N}}w_{i}(x)>0$. $M_{i}$ are bounded continuous
functions, $F$ belongs to $C^{1}(\mathbb{R}^{N}\times\mathbb{R}^{n})$ and
satisfies adequate growth assumptions and $F_{u_{i}}$ denotes the partial
derivative of $F$ with respect to $u_{i}$ for all $1\leq i\leq n$. The
functions $p_{i}(x),q_{i}(x)$ and $\gamma_{i}(x)$ are continuous real-valued
functions such that

$\displaystyle 1<p_{i}^{-}\leq p_{i}(x)\leq p_{i}^{+}<q_{i}^{-}\leq
q_{i}(x)\leq q_{i}^{+}<N,$ (1.2)

and

$\gamma_{i}(x)=(1-\mathcal{H}(k^{3}_{i}))p_{i}(x)+\mathcal{H}(k^{3}_{i})q_{i}(x),$

for all $x\in\mathbb{R}^{N}$ with $k^{3}_{i}$ is given in
$(\textbf{{H}}_{2})$, where $\displaystyle
p_{i}^{-}:=\inf_{x\in\mathbb{R}^{N}}p_{i}(x)$, $\displaystyle
p_{i}^{+}:=\sup_{x\in\mathbb{R}^{N}}p_{i}(x)$, and analogously to
$q_{i}^{-},q_{i}^{+},\gamma_{i}^{-}$ and $\gamma_{i}^{+}$, with
$\displaystyle\mathcal{H}:\mathbb{R}_{0}^{+}\to\left\\{0,1\right\\}$ is given
by

$\mathcal{H}(k_{i})=\begin{cases}1&\text{ if }k_{i}\geq 0,\\\ 0&\text{ if
}k_{i}<0.\end{cases}$

The operator $\mathcal{A}_{ji}:X_{i}\to\mathbb{R}^{n}$ with $j=1$ or $2$, and
the operator $\mathcal{B}_{i}:X_{i}\to\mathbb{R}$, are respectively defined by

$\mathcal{A}_{ji}(u_{i})=a_{ji}(|u_{i}|^{p_{i}(x)})|u_{i}|^{p_{i}(x)-2}u_{i},\text{
and
}\mathcal{B}_{i}(u_{i})=\displaystyle\int_{\mathbb{R}^{N}}\dfrac{1}{p_{i}(x)}\Big{(}A_{1i}(|\nabla
u_{i}|^{p_{i}(x)})+w_{i}(x)A_{2i}(|u_{i}|^{p_{i}(x)})\Big{)}dx,$ (1.3)

where $X_{i}$ is the Banach space
$X_{i}:=W_{w_{i}}^{1,p_{i}(x)}(\mathbb{R}^{N})\cap
W_{w_{i}}^{1,\gamma_{i}(x)}(\mathbb{R}^{N})$, $A_{ji}(.)$ is the function
$A_{ji}(t)=\displaystyle\int_{0}^{t}a_{ji}(k)dk$, and the function $a_{i}(.)$
is described in the hypothesis $(\textbf{{H}}_{1})$.

In this article, we consider the function
$a_{ji}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying the following hypotheses
for all $1\leq i\leq n$ and $j\in\\{1,2\\}$:

  * $(\textbf{{H}}_{1})$

The function $a_{ji}(.)$ is of class $C^{1}$.

  * $(\textbf{{H}}_{2})$

There exist positive constants $k_{ji}^{0},k_{ji}^{1},k_{ji}^{2}$ and
$k_{i}^{3}$ for all $1\leq i\leq n$, such that

$k_{ji}^{0}+\mathcal{H}(k_{i}^{3})k_{ji}^{2}\tau^{\frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}}\leq
a_{ji}(\tau)\leq
k_{ji}^{1}+k_{i}^{3}\tau^{\frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}},$

for all $\tau\geq 0$ and for almost every $x\in\mathbb{R}^{N}.$

  * $(\textbf{{H}}_{3})$

There exist $c_{ji}>0$ such that

$\min\left\\{a_{ji}(\tau^{p_{i}(x)})\tau^{p_{i}(x)-2},a_{ji}(\tau^{p_{i}(x)})\tau^{p_{i}(x)-2}+\tau\frac{\partial(a_{ji}(\tau^{p_{i}(x)})\tau^{p_{i}(x)-2})}{\partial\tau}\right\\}\geq
c_{ji}\tau^{p_{i}(x)-2},$

for almost every $x\in\mathbb{R}^{N}$ and for all $\tau>0$.

  * $(\textbf{{H}}_{4})$

There exists positive constants $\beta_{ji}$ for all
$i\in\left\\{1,2,\dots,n\right\\}$ such that

$A_{ji}(\tau)\geq\frac{1}{\beta_{ji}}a_{ji}(\tau)\tau,$

for all $\tau\geq 0$.

  * $(\textbf{{M}})$

$M_{i}:\mathbb{R}^{+}\longrightarrow\mathbb{R}$ are continuous and increasing
functions such that $0<m_{0}=\displaystyle\min_{1\leq i\leq n}M_{i}(0)\leq
M_{i}(t)\leq m_{1}=\max_{1\leq i\leq n}M_{i}(t)$, for all $t\geq 0$,
$i\in\left\\{1,2,\dots,n\right\\}$.

Recently, a great attention has been focused on the study of this type of
problems. They appear in mathematical models in different branches in science
as electrorheological fluids [30, 31], elastic mechanics [34], stationary
thermorheological viscous flows of non-Newtonian fluids [29], image processing
[13], and mathematical description of the processes filtration of barotropic
gas through a porous medium [3].

The system (1.1) is related (in the case of a single equation) to a model
firstly proposed by Kirchhoff in 1883 as the stationary version of the
Kirchhoff equation

$\rho\frac{\partial^{2}u}{\partial
t^{2}}-\left(\dfrac{\rho_{0}}{h}+\dfrac{E}{2L}\int_{0}^{L}\left|\dfrac{\partial
u(x)}{\partial x}\right|^{2}dx\right)\dfrac{\partial^{2}u}{\partial x^{2}}=0,$
(1.4)

where $\rho$, $\rho_{0}$, $E$ and $L$ are constants. This equation extends the
classical D’Alembert’s wave equation by considering the effects of the changes
in the length of the strings during the vibrations. A distinguishing feature
of equation (1.4) is that the equation contains a nonlocal coefficient
$\dfrac{\rho_{0}}{h}+\dfrac{E}{2L}{\displaystyle\int_{0}^{L}\left|\dfrac{\partial
u}{\partial x}\right|^{2}dx}$ which depends on the average
$\dfrac{1}{L}{\displaystyle\int_{0}^{L}\left|\dfrac{\partial u}{\partial
x}\right|^{2}dx}$, and hence the equation is no longer a pointwise equation.
The parameters in equation (1.4) have the following meanings: $E$ is the Young
modulus of the material, $\rho$ is the mass density, $L$ is the length of the
string, $h$ is the area of cross-section, and $\rho_{0}$ is the initial
tension (see [23]).

Now, in order to illustrate the degree of generality of the kind of problems
studied here, with adequate hypotheses on the functions $a_{ji}$, in the
following we present more some examples of problems which are also interesting
from the mathematical point of view and have a wide range of applications in
physics and related sciences.

Example I. Considering $a_{ji}\equiv 1$, we have that $a_{ji}$ satisfies the
$(\textbf{{H}}_{1}),(\textbf{{H}}_{2})$ and $(\textbf{{H}}_{3})$ with
$k^{0}_{ji}=k^{1}_{ji}=1$ and $k^{2}_{ji}>0$ and $k^{3}_{i}=0$ for all
$i\in\\{1,2,...,N\\}$ and for $j=1$ or $2$. In this case we are studying
problem:

$\displaystyle\begin{cases}-M_{i}\Big{(}\mathcal{B}_{i}(u_{i})\Big{)}\Big{(}\Delta_{p_{i}(x)}u_{i}-w_{i}(x)|u_{i}|^{p_{i}(x)-2}u_{i}\Big{)}=\lambda
F_{u_{i}}(x,u)&\text{ in }\mathbb{R}^{N},\\\ \quad u_{i}\in
W_{w_{i}}^{1,p_{i}(x)}(\mathbb{R}^{N})\\\ \end{cases}$ (1.5)

where
$\mathcal{B}_{i}(u_{i})=\displaystyle\int_{\mathbb{R}^{N}}\dfrac{1}{p_{i}(x)}\left(|\nabla
u_{i}|^{p_{i}(x)}+w_{i}(x)|u_{i}|^{p_{i}(x)}\right)dx$. The operator
$\Delta_{p_{i}(x)}u_{i}:=\textrm{div}\,(|\nabla u_{i}|^{p_{i}(x)-2}\nabla
u_{i})$ is so-called $p_{i}(x)$-Laplacian, which coincides with the usual
$p_{i}$-Laplacian when $p_{i}(x)=p_{i}$, and with the Laplacian when
$p_{i}(x)=2$.

Example II. Considering $a_{ji}(t)=1+t^{\frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}}$,
we have that $a_{ji}$ satisfies the $(\textbf{{H}}_{1}),(\textbf{{H}}_{2})$
and $(\textbf{{H}}_{3})$ with
$k^{0}_{ji}=k^{1}_{ji}=k^{2}_{ji}=\kappa_{i}^{3}=1$ for all
$i\in\\{1,2,...,n\\}$ and for $j=1$ or $2$). In this case we are studying the
$p_{i}\&q_{i}$-Laplacian equation:

$\displaystyle\begin{cases}-M_{i}\Big{(}\mathcal{B}_{i}(u_{i})\Big{)}\Big{(}\Delta_{p_{i}(x)}u_{i}+\Delta_{q_{i}(x)}u_{i}-w_{i}(x)(|u_{i}|^{p_{i}(x)-2}u_{i}+|u_{i}|^{q_{i}(x)-2}u_{i})\Big{)}=\lambda
F_{u_{i}}(x,u)&\text{ in }\mathbb{R}^{N},\\\ \qquad u_{i}\in
W_{w_{i}}^{1,p_{i}(x)}(\mathbb{R}^{N})\cap
W_{w_{i}}^{1,q_{i}(x)}(\mathbb{R}^{N})\\\ \end{cases}$ (1.6)

where
$\mathcal{B}_{i}(u_{i})=\displaystyle\int_{\mathbb{R}^{N}}\Bigg{(}\dfrac{1}{p_{i}(x)}\left(|\nabla
u_{i}|^{p_{i}(x)}+w_{i}(x)|u_{i}|^{p_{i}(x)}\right)+\dfrac{1}{q_{i}(x)}\left(|\nabla
u_{i}|^{q_{i}(x)}+w_{i}(x)|u_{i}|^{q_{i}(x)}\right)\Bigg{)}dx$.

This class of problems comes, for example, from a general reaction-diffusion
system

$u_{t}=\textrm{div}[D(u)\nabla u]+h(x,u),$ (1.7)

where $D(u)=|\nabla u|^{p(x)-2}+|\nabla u|^{q(x)-2}$, and the reaction term
$h(x,u)$ is a polynomial of u with variable coefficients. This system has a
wide range of applications in physics and related sciences, such as
biophysics, plasma physics and chemical reaction design. In such applications,
the function $u$ describes a concentration, the first term on the right-hand
side of (1.7) corresponds to the diffusion with a diffusion coefficient
$D(u)$; whereas the second one is the reaction and relates to source and loss
processes. Typically, in chemical and biological applications, (for further
details, see [26, 21] references therein).

We continued with other examples that are also interesting from mathematical
point of view:

Example III. Considering $a_{1i}(t)=1+\frac{t}{\sqrt{1+t^{2}}}$ and
$a_{2i}\equiv 1$, we have that $a_{1i}$ and $a_{2i}$ satisfies the
$(\textbf{{H}}_{1}),(\textbf{{H}}_{2})$ and $(\textbf{{H}}_{3})$ with
$k^{0}_{1i}=k^{0}_{2i}=k^{1}_{2i}=1$,$k^{1}_{1i}=2$, and $k^{3}_{i}=0$,
$k^{2}_{1i}>0$ and $k^{2}_{2i}>0$. In this case we are studying problem:

$\displaystyle\begin{cases}-M_{i}\Big{(}\mathcal{B}_{i}(u_{i})\Big{)}\Big{(}\textrm{div}\,\Big{(}\Big{(}1+\frac{|\nabla
u_{i}|^{p_{i}(x)}}{\sqrt{1+|\nabla u_{i}|^{2p_{i}(x)}}}\Big{)}|\nabla
u_{i}|^{p_{i}(x)-2}\nabla
u_{i}\Big{)}-w_{i}(x)|u_{i}|^{p_{i}(x)-2}u_{i}\Big{)}=\lambda
F_{u_{i}}(x,u)&\text{ in }\mathbb{R}^{N},\\\ \quad u_{i}\in
W_{w_{i}}^{1,p_{i}(x)}(\mathbb{R}^{N}),\end{cases}$ (1.8)

where
$\mathcal{B}_{i}(u_{i})=\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{p_{i}(x)}\Big{(}|\nabla
u_{i}|^{p_{i}(x)}+\sqrt{1+|\nabla
u_{i}|^{2p_{i}(x)}}+w_{i}(x)|u_{i}|^{p_{i}(x)}\Big{)}dx$.

The operator $\textrm{div}\,\Big{(}\Big{(}1+\frac{|\nabla
u|^{p(x)}}{\sqrt{1+|\nabla u|^{2p(x)}}}\Big{)}|\nabla u|^{p(x)-2}\nabla
u\Big{)}$ is so-called $p(x)$-Laplacian like or so-called the generalized
Capillary operator. The Capillarity can be briefly explained by considering
the effects of two opposing forces: adhesion, i.e. the attractive (or
repulsive) force between the molecules of the liquid and those of the
container; and cohesion, i.e. the attractive force between the molecules of
the liquid. The study of capillary phenomenon has gained much attention. This
increasing interest is motivated not only by the fascination in naturally-
occurring phenomena such as motion of drops, bubbles and waves, but also its
importance in applied fields ranging from industrial and biomedical and
pharmaceutical to microfluidic systems see [27].

Example IV. Considering $\displaystyle
a_{1i}(t)=1+t^{\frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}}+\frac{1}{(1+t)^{\frac{p_{i}(x)-2}{p_{i}(x)}}}$
and $\displaystyle a_{2i}(t)=1+t^{\frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}}$, we
have that $\displaystyle a_{1i}$ and $\displaystyle a_{2i}$ satisfies the
$\displaystyle(\textbf{{H}}_{1}),(\textbf{{H}}_{2})$ and $(\textbf{{H}}_{3})$
with $k^{0}_{1i}=k^{0}_{2i}=k^{1}_{2i}=1$, $k^{1}_{1i}=2$ and
$k^{3}_{i}=k^{2}_{1i}=k^{2}_{2i}=1$. In this case we are studying problem

$\displaystyle\begin{cases}-M_{i}\Big{(}\mathcal{B}_{i}(u_{i})\Big{)}\Bigg{[}\Delta_{p_{i}(x)}u_{i}+\Delta_{q_{i}(x)}u_{i}+\textrm{div}\,\Big{(}\dfrac{|\nabla
u_{i}|^{p_{i}(x)-2}\nabla u_{i}}{(1+|\nabla
u_{i}|^{p_{i}(x)})^{\frac{p_{i}(x)-2}{p_{i}(x)}}}\Big{)}-w_{i}(x)\Big{(}|u_{i}|^{p_{i}(x)-2}u_{i}+|u_{i}|^{q_{i}(x)-2}u_{i}\Big{)}\Bigg{]}=\lambda
F_{u_{i}}(x,u)&\text{ in }\mathbb{R}^{N},\\\ \qquad u_{i}\in
W_{w_{i}}^{1,p_{i}(x)}(\mathbb{R}^{N})\cap
W_{w_{i}}^{1,q_{i}(x)}(\mathbb{R}^{N}),\end{cases}$ (1.9)

where
$\mathcal{B}_{i}(u_{i})=\displaystyle\int_{\mathbb{R}^{N}}\Bigg{(}\dfrac{1}{p_{i}(x)}\left(|\nabla
u_{i}|^{p_{i}(x)}+w_{i}(x)|u_{i}|^{p_{i}(x)}\right)+\dfrac{1}{q_{i}(x)}\left(|\nabla
u_{i}|^{q_{i}(x)}+w_{i}(x)|u_{i}|^{q_{i}(x)}\right)+\frac{1}{2}(1+|\nabla
u_{i}|^{p_{i}(x)})^{\frac{2}{p_{i}(x)}}\Bigg{)}dx$.

In the literature, the existence and multiplicity of the solutions for
quasilinear elliptic systems have been studied by many authors ([1, 6, 8, 9,
10, 15, 16, 25, 32]), where the general variable exponent elliptic operators
$\mathcal{A}_{ji}$ in divergence form and the nonlinear potential $F$ have
different mixed growth conditions. For example in [18] the authors show the
existence of nontrivial solutions for the following $(p,q)$-Laplacian system

$\displaystyle\begin{cases}\displaystyle-\Delta_{p}u=\frac{\partial
F}{\partial u}(x,u,v)\quad\text{in }\mathbb{R}^{N}\\\ &\\\
\displaystyle-\Delta_{q}v=\frac{\partial F}{\partial v}(x,u,v)\quad\text{in
}\mathbb{R}^{N}\end{cases}$ (1.10)

where the potential function $F$ satisfies mixed and subcritical growth
conditions and, in addition, is supposed to be intimately connected with the
first eigenvalue of the $(\Delta_{p},\Delta_{q})$-operator. They apply the
Mountain Pass theorem to get existence of nontrivial solutions. In [11], the
authors using an abstract critical point result of Bonanno and Marano
established the existence of an interval $\Lambda\subseteq[0,+\infty[$ such
that for each $\lambda\in\Lambda$ the quasilinear elliptic system

$-\Delta_{p_{i}(x)}u_{i}+a_{i}(x)|u_{i}|^{p_{i}(x)-2}u_{i}=\lambda
F_{u_{i}}(x,u_{1},u_{2},...,u_{n})\quad\text{in }\mathbb{R}^{N},$ (1.11)

for $1\leq i\leq n$, where $a_{i}\in L^{\infty}(\mathbb{R}^{N})$ such that
$\displaystyle a_{i}:=ess\inf_{x\in\mathbb{R}^{N}}a_{i}(x)>0$ and $\lambda$ is
a positive parameter, has at least three distinct nontrivial solutions, and in
[12] some similar results were obtained for the following classe of nonlocal
elliptic systems

$\displaystyle\begin{cases}-M_{1}\left({\displaystyle\int_{\mathbb{R}^{N}}\dfrac{1}{p(x)}\left(|\nabla
u|^{p(x)}+a(x)|u|^{p(x)}\right)dx}\right)\big{(}\Delta_{p(x)}u-a(x)|u|^{p(x)-2}u\big{)}=\lambda
F_{u}(x,u,v)&\quad\text{in }\mathbb{R}^{N},\\\
-M_{2}\left({\displaystyle\int_{\mathbb{R}^{N}}\dfrac{1}{q(x)}\left(|\nabla
v|^{q(x)}+a(x)|v|^{q(x)}\right)dx}\right)\big{(}\Delta_{q(x)}v-b(x)|v|^{q(x)-2}v\big{)}=\lambda
F_{v}(x,u,v)&\quad\text{in }\mathbb{R}^{N};\end{cases}$ (1.12)

where $a$, $b\in L^{\infty}(\mathbb{R}^{N})$ such that $\displaystyle
a:=ess\inf_{x\in\mathbb{R}^{N}}a(x)>0$ and $\displaystyle
b:=ess\inf_{x\in\mathbb{R}^{N}}b(x)>0$. $M_{1}$ and $M_{2}$ are bounded
continuous functions, $F$ belongs to
$C^{1}(\mathbb{R}^{N}\times\mathbb{R}^{2})$ and verifies some mixed growth
conditions.

This paper’s primary purpose is to establish the existence of some interval
which includes $\lambda$, where the system (1.1) admits at least three weak
solutions, which extend, complement and complete in several ways some of many
works in particular the results in [11, 12], by means of an abstract critical
points result of G. Bonanno and S.A. Marano [7], which is a more precise
version of Theorem 3.2 of [6]. For other basic notations and definitions we
refer to [33].

Now, and for the convenience of the readers, we recall the three critical
points theorem of G. Bonanno and S.A. Marano [7] which is our main tool to
prove the results. Here, $X^{*}$ denotes the dual space of $X$.

######  Lemma 1.1 (see [7, Theorem 3.6]).

Let $X$ be a reflexive real Banach space; $\Phi:X\rightarrow\mathbb{R}$ be a
coercive, continuously Gâteaux differentiable and sequentially weakly lower
semicontinuous functional whose Gâteaux derivative admits a continuous inverse
on $X^{*}$; $\Psi:X\rightarrow\mathbb{R}$ be a continuously Gâteaux
differentiable functional whose Gâteaux derivative is compact such that

$\Phi(0)=\Psi(0)=0.$

Assume that there exist $r>0$ and $\overline{x}\in X,$ with
$r<\Phi(\overline{x}),$ such that

  1. $\rm(a_{1})$

$\displaystyle\frac{\displaystyle\sup_{\Phi(\bar{x})\leq
r}\Psi(\bar{x})}{r}<\frac{\Psi(\overline{x})}{\Phi(\overline{x})}$;

  2. $\rm(a_{2})$

for each
$\displaystyle\lambda\in\Lambda_{r}:=]\frac{\Phi(\overline{x})}{\Psi(\overline{x})},\frac{r}{\displaystyle\sup_{\Phi(\bar{x})\leq
r}\Psi(\bar{x})}[$ the functional $\Phi-\lambda\Psi$ is coercive.

Then, for each $\lambda\in\Lambda_{r}$ the functional $\Phi-\lambda\Psi$ has
at least three distinct critical points in $X$.

Organization of the paper. The rest of the paper is organized as follows:
Section 2 contains some basic preliminary knowledge of the variable exponent
spaces and some results which will be needed later. Finally, in Section 3, we
state and establish our main result.

##  2\. Preliminaries and basic notations

In this section, we introduce some definitions and results which will be used
in the next section. Firstly, we introduce some theories of Lebesgue-Sobolev
spaces with variable exponent. The details can be found in [17, 20, 24, 28].
Denote by $S(\mathbb{R}^{N})$ the set of all measurable real functions on
$\mathbb{R}^{N}$. Set

$C_{+}(\mathbb{R}^{N})=\\{p\in
C(\mathbb{R}^{N}):\inf_{x\in\mathbb{R}^{N}}p(x)>1\\}.$

For any $p\in C_{+}(\mathbb{R}^{N})$ we define

$\displaystyle p^{-}:=\inf_{x\in\mathbb{R}^{N}}p(x)\quad\text{
and}~{}p^{+}:=\sup_{x\in\mathbb{R}^{N}}p(x).$

For any $p\in C_{+}(\mathbb{R}^{N})$, we define the variable exponent Lebesgue
space as

$\displaystyle L^{p(x)}(\mathbb{R}^{N})=\left\\{u\in
S(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}|u(x)|^{p(x)}dx<\infty\right\\},$

endowed with the Luxemburg norm

$|u|_{p(x)}:=|u|_{L^{p(x)}(\mathbb{R}^{N})}=\inf\left\\{\mu>0:\int_{\mathbb{R}^{N}}\left|\frac{u(x)}{\mu}\right|^{p(x)}dx\leq
1\right\\}.$

Let $w\in S(\mathbb{R}^{N})$, and $w(x)>0$ for a.e $x\in\mathbb{R}^{N}$.
Define the weighted variable exponent Lebesgue space
$L_{w}^{p(x)}(\mathbb{R}^{N})$ by

$\displaystyle L_{w}^{p(x)}(\mathbb{R}^{N})=\left\\{u\in
S(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}w(x)|u(x)|^{p(x)}dx<\infty\right\\},$

with the norm

$|u|_{p(x),w(x)}:=|u|_{L_{w}^{p(x)}(\mathbb{R}^{N})}=\inf\left\\{\mu>0:\int_{\mathbb{R}^{N}}w(x)\left|\frac{u(x)}{\mu}\right|^{p(x)}dx\leq
1\right\\}.$

From now on, we suppose that $w\in L^{\infty}(\mathbb{R}^{N})$ with
$\displaystyle w:=ess\inf_{x\in\mathbb{R}^{N}}w(x)>0$. Then obviously
$L_{w}^{p(x)}(\Omega)$ is a Banach space (see [14] for details).

On the other hand, the variable exponent Sobolev space
$W^{1,p(x)}(\mathbb{R}^{N})$ is defined by

$W^{1,p(x)}(\mathbb{R}^{N})=\\{u\in L^{p(x)}(\mathbb{R}^{N}):|\nabla u|\in
L^{p(x)}(\mathbb{R}^{N})\\},$

and is endowed with the norm

$\|u\|_{1,p(x)}:=\|u\|_{W^{1,p(x)}(\mathbb{R}^{N})}=|u|_{p(x)}+|\nabla
u|_{p(x)},~{}~{}~{}~{}\forall u\in W^{1,p(x)}(\mathbb{R}^{N}).$

Next, the weighted-variable exponent Sobolev space
$W_{w}^{1,p(x)}(\mathbb{R}^{N})$ is defined by

$W_{w}^{1,p(x)}(\mathbb{R}^{N})=\\{u\in L_{w}^{p(x)}(\mathbb{R}^{N}):|\nabla
u|\in L_{w}^{p(x)}(\mathbb{R}^{N})\\},$

with the norm

$\|u\|_{p(x),w}:=\inf\left\\{\mu>0:\int_{\mathbb{R}^{N}}\left|\frac{\nabla
u(x)}{\mu}\right|^{p(x)}+w(x)\left|\frac{u(x)}{\mu}\right|^{p(x)}dx\leq
1\right\\},\forall u\in W_{w}^{1,p(x)}(\mathbb{R}^{N}).$

Then the norms $\|u\|_{p(x),w}$ and $\|u\|_{p(x)}$ are equivalent in
$W_{w}^{1,p(x)}(\mathbb{R}^{N})$.

If $p^{-}>1$, then the spaces $L^{p(x)}(\mathbb{R}^{N})$,
$W^{1,p(x)}(\mathbb{R}^{N})$ and $W_{w}^{1,p(x)}(\mathbb{R}^{N})$ are
separable, reflexive and uniformly convex Banach spaces.

Here we display some facts which will be used later.

######  Proposition 2.1 (see [17, 20]).

The topological dual space of $L^{p(x)}(\Omega)$ is
$L^{p^{\prime}(x)}(\Omega)$, where

$\frac{1}{p(x)}+\frac{1}{p^{\prime}(x)}=1.$

Moreover, for any $(u,v)\in L^{p(x)}(\Omega)\times L^{p^{\prime}(x)}(\Omega)$,
we have

$\Big{|}\int_{\Omega}uvdx\Big{|}\leq(\frac{1}{p^{-}}+\frac{1}{(p^{\prime})^{-}})|u|_{p(x)}|v|_{p^{\prime}(x)}\leq
2|u|_{p(x)}|v|_{p^{\prime}(x)}.$

######  Proposition 2.2 (see [17, 20]).

Denote $\displaystyle\rho_{p(x)}(u):=\int_{\mathbb{R}^{N}}|u|^{p(x)}dx$, for
all $u\in L^{p(x)}(\mathbb{R}^{N})$. We have

$\min\\{|u|_{p(x)}^{p^{-}},|u|_{p(x)}^{p^{+}}\\}\leq\rho_{p(x)}(u)\leq\max\\{|u|_{p(x)}^{p^{-}},|u|_{p(x)}^{p^{+}}\\},$

and the following implications are true

  * (i)

$|u|_{p(x)}<1$ (resp. $=1,>1$) $\Leftrightarrow\rho_{p(x)}(u)<1$ (resp.
$=1,>1$),

  * (ii)

$|u|_{p(x)}>1\Rightarrow|u|_{p(x)}^{p^{-}}\leq\rho_{p(x)}(u)\leq|u|_{p(x)}^{p^{+}}$,

  * (iii)

$|u|_{p(x)}<1\Rightarrow|u|_{p(x)}^{p^{+}}\leq\rho_{p(x)}(u)\leq|u|_{p(x)}^{p^{-}}$.

Denote $\displaystyle\rho_{p(x),w}(u):=\int_{\mathbb{R}^{N}}\left(\left|\nabla
u(x)\right|^{p(x)}+w(x)\left|u(x)\right|^{p(x)}\right)dx$, for all $u\in
W_{w}^{1,p(x)}(\mathbb{R}^{N})$. From Proposition (2.2), we have

$\|u\|_{p(x),w}^{p^{-}}\leq\rho_{p(x),w}(u)\leq\|u\|_{p(x),w}^{p^{+}}\quad\text{
if}~{}~{}~{}~{}\|u\|_{p(x),w}\geq 1,$ (2.1)
$\|u\|_{p(x),w}^{p^{+}}\leq\rho_{p(x),w}(u)\leq\|u\|_{p(x),w}^{p^{-}}\quad\text{
if}~{}~{}~{}~{}\|u\|_{p(x),w}\leq 1.$ (2.2)

######  Proposition 2.3 (see [19]).

Let $p(x)$ and $q(x)$ be measurable functions such that $p\in
L^{\infty}(\mathbb{R}^{N})$ and $1\leq p(x),q(x)<\infty$ almost everywhere in
$\mathbb{R}^{N}$. If $u\in L^{q(x)}(\mathbb{R}^{N})$, $u\neq 0$, then we have

$\displaystyle|u|_{p(x)q(x)}\leq
1\Rightarrow|u|_{p(x)q(x)}^{p^{-}}\leq\big{|}|u|^{p(x)}\big{|}_{q(x)}\leq|u|_{p(x)q(x)}^{p^{+}},$
$\displaystyle|u|_{p(x)q(x)}\geq
1\Rightarrow|u|_{p(x)q(x)}^{p^{+}}\leq\big{|}|u|^{p(x)}\big{|}_{q(x)}\leq|u|_{p(x)q(x)}^{p^{-}}.$

In particular, if $p(x)=p$ is constant, then

$||u|^{p}|_{q(x)}=|u|_{pq(x)}^{p}.$

Now, for all $x\in\mathbb{R}^{N}$, denote

$p^{\partial}(x)=\begin{cases}\frac{Np(x)}{N-p(x)}&\text{for }p(x)<N\\\
+\infty&\text{for }p(x)\geq N\end{cases}$

the critical Sobolev exponent of $p(x)$. We have

######  Proposition 2.4 (see [17, 19]).

Let $r\in C_{+}^{0,1}(\mathbb{R}^{N})$, the space of Lipschitz-continuous
functions defined on $\mathbb{R}^{N}$. Then, there exists a positive constant
$c_{r}$ depending on $r$ such that

$|u|_{r^{\partial}(x)}\leq c_{r}\|u\|_{r(x),w},\quad\forall u\in
W_{w}^{1,r(x)}(\mathbb{R}^{N}).$

######  Proposition 2.5 (see [17, 19]).

Assume that $p\in C(\mathbb{R}^{N})$ with $p(x)>1$ for each
$x\in\mathbb{R}^{N}$. If $r\in C(\mathbb{R}^{N})$ is such that
$1<r(x)<p^{\partial}(x)$ for each $x\in\Omega$, then there exists a continuous
and compact embedding $W^{1,p(x)}(\mathbb{R}^{N})\hookrightarrow
L^{r(x)}(\mathbb{R}^{N})$.

In the following, we shall use the product space

$X:=\prod_{i=1}^{n}\Big{(}W_{w_{i}}^{1,p_{i}(x)}(\mathbb{R}^{N})\cap
W_{w_{i}}^{1,\gamma_{i}(x)}(\mathbb{R}^{N})\Big{)},$

equipped with the norm

$\|u\|:=\max\left\\{\|u_{i}\|_{w_{i}}\right\\},~{}~{}~{}~{}~{}\forall
u=(u_{1},\dots,u_{n})\in X,$

where
$\|u_{i}\|_{w_{i}}:=\|u_{i}\|_{p_{i}(x),w_{i}}+\mathcal{H}(\kappa_{i}^{3})\|u_{i}\|_{q_{i}(x),w_{i}}$
is the norm in $W_{w_{i}}^{1,p_{i}(x)}(\mathbb{R}^{N})\cap
W_{w_{i}}^{1,\gamma_{i}(x)}(\mathbb{R}^{N})$. We denote $X^{\star}$ the
topological dual of $X$ equipped with the usual dual norm.

######  Definition 2.6.

Let $X$ be a Banach space, an element $u=(u_{1},u_{2},...,u_{n})\in X$ is
called a weak solution of the system (1.1) if

$\displaystyle\sum_{i=1}^{n}M_{i}\left(\mathcal{B}_{i}(u_{i})\right)\int_{\mathbb{R}^{N}}\Big{(}\mathcal{A}_{1i}(\nabla
u_{i})\nabla
v_{i}+w_{i}(x)\mathcal{A}_{2i}(u_{i})v_{i}\Big{)}\,dx-\sum_{i=1}^{n}\displaystyle\int_{\Omega}\lambda
F_{u_{i}}(x,u_{1},...u_{n})v_{i}\,dx=0,$

for all $\displaystyle v=(v_{1},v_{2},...,v_{n})\in
X=\prod_{i=1}^{n}(W_{w_{i}}^{1,p_{i}(x)}(\mathbb{R}^{N})\cap
W_{w_{i}}^{1,\gamma_{i}(x)}(\mathbb{R}^{N}))$.

For every $u=(u_{1},\dots,u_{n})$ in $X$, let us define the functional $\Psi$
by

$\displaystyle\Psi(u)=\int_{\mathbb{R}^{N}}F(x,u_{1},\dots,u_{n})dx$.

Under the assumptions below $(\mathcal{F}_{1})$ and $(\mathcal{F}_{2})$, we
have $\displaystyle\Psi\in C^{1}(X,\mathbb{R})$ and its derivative is given by

$\displaystyle\Psi^{\prime}(u)v=\sum_{i=1}^{n}D_{i}\Psi(u)v_{i}$,

where

$\displaystyle D_{i}\Psi(u)v_{i}=\int_{\mathbb{R}^{N}}\frac{\partial
F}{\partial u_{i}}(x,u_{1},\dots,u_{n})v_{i}dx$,

for all $\displaystyle v=(v_{1},v_{2},...,v_{n})\in X$. Define also $\Phi$ in
$X$ by

$\displaystyle\Phi(u)=\sum_{i=1}^{n}\Phi_{i}(u_{i})$,

where

$\Phi_{i}(u_{i})=\widehat{M_{i}}\left(\mathcal{B}_{i}(u_{i}(x))\right),$

for any $u=(u_{1},\dots,u_{n})$ in $X$, with
$\displaystyle\widehat{M}_{i}(t):=\int_{0}^{t}M_{i}(s)ds$ for all $t\geq 0$,
($i\in\\{1,\dots,n\\}$).

We recall that $\Phi$ is a $C^{1}$-functional, weakly lower semi-continuous
and its derivative is given by

$\displaystyle\Phi^{\prime}(u)v=\sum_{i=1}^{n}D_{i}\Phi(u)v_{i}$,

where

$\displaystyle
D_{i}\Phi(u)v_{i}=M_{i}\left(\mathcal{B}_{i}(u_{i})\right)\int_{\mathbb{R}^{N}}\Big{(}\mathcal{A}_{1i}(\nabla
u_{i})\nabla v_{i}+w_{i}(x)\mathcal{A}_{2i}(u_{i})v_{i}\Big{)}\,dx$,

for all $v=(v_{1},\dots,v_{n})\in X$.

The Euler-Lagrange functional associated to the system (1.1) is defined by

$\displaystyle E_{\lambda}(u):=\Phi(u)-\lambda\Psi(u)\quad\forall u\in X$.

Since $E_{\lambda}\in C^{1}(X,\mathbb{R})$ and for all
$v=(v_{1},\dots,v_{n})\in X$, we have

$\displaystyle
E_{\lambda}^{\prime}(u)v=\Phi^{\prime}(u)v-\lambda\Psi^{\prime}(u)v$

Consequently, $u\in X$ is a weak solution of (1.1) if and only if $u$ is a
critical point of $E_{\lambda}$.

We end this section by stating some hypotheses that we will be using later.

### Hypotheses

We assume some growth conditions:

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

$F\in C^{1}(\mathbb{R}^{N}\times\mathbb{R}^{n},\mathbb{R})$ and
$F(x,0,\dots,0)=0$.

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

There exist positive functions $b_{ij}$ ($1\leq i,j\leq n$), such that

$\displaystyle\Big{|}\frac{\partial F}{\partial
u_{i}}(x,u_{1},...,u_{n})\Big{|}\leq\sum_{j=1}^{n}b_{ij}(x)|u_{j}|^{\mu_{ij}-1},$

where $\displaystyle 1<\mu_{ij}<\inf_{x\in\mathbb{R}^{N}}\gamma_{i}(x)$ for
all $x\in\mathbb{R}^{N}$ and for all $i\in\left\\{1,2,...,n\right\\}$. The
weight-functions $b_{ii}$ (resp $b_{ij}$ if $i\neq j$) belong to the
generalized Lebesgue spaces $L^{\alpha_{i}}(\mathbb{R}^{N})$ (resp
$L^{\alpha_{ij}}(\mathbb{R}^{N})$), with

$\alpha_{i}(x)=\frac{\gamma_{i}(x)}{\gamma_{i}(x)-1},\quad\alpha_{ij}(x)=\frac{\gamma_{i}^{\partial}(x)\gamma_{j}^{\partial}(x)}{\gamma_{i}^{\partial}(x)\gamma_{j}^{\partial}(x)-\gamma_{i}^{\partial}(x)-\gamma_{j}^{\partial}(x)}.$

Example: We give an example of potential $F$ satisfying hypotheses
$(\mathcal{F}_{1})$ and $(\mathcal{F}_{2})$ for $n=2$. Let

$F(x,u_{1},u_{2})=a(x)|u_{1}|^{l_{1}(x)}|u_{2}|^{l_{2}(x)},$

where
$\displaystyle\frac{l_{1}(x)}{\gamma_{1}(x)}+\frac{l_{2}(x)}{\gamma_{2}(x)}<1$
and $a$ is a positive function in $L^{e(x)}(\mathbb{R}^{N})$ such that

$\displaystyle
e(x)=\dfrac{\gamma_{1}^{\partial}(x)\gamma_{2}^{\partial}(x)}{\gamma_{1}^{\partial}(x)\gamma_{2}^{\partial}(x)-l_{1}(x)\gamma_{2}^{\partial}(x)-l_{2}(x)\gamma_{2}^{\partial}(x)}$,

for each $x\in\mathbb{R}^{N}$.

We can easily verify that $F(x,u_{1},u_{2})$ satisfies the condition
$(\mathcal{F}_{1})$. Moreover, by using Young inequality we easily check that
the condition $(\mathcal{F}_{2})$.

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

Assume that there exist $r>0$ and $z=(z_{1},...,z_{n})\in X$ such that the
following conditions are satisfied:

    * $(C_{1})$

$\displaystyle\sum_{i=1}^{n}\frac{m_{0}^{\star}}{p_{i}^{+}}\Big{(}\min\left\\{\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{-}},\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{+}}\right\\}+\mathcal{H}(k_{i}^{3})\min\left\\{\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{-}},\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{+}}\right\\}\Big{)}>r.$

    * $(C_{2})$

$\displaystyle\frac{\displaystyle\int_{\mathbb{R}^{N}}\sup_{(\xi_{1},\dots,\xi_{n})\in
K(\frac{sr}{m_{0}^{\star}})}F(x,\xi_{1},\dots,\xi_{n})dx}{r}<\dfrac{\displaystyle\int_{\mathbb{R}^{N}}F(x,z_{1},\dots,z_{n})dx}{\displaystyle
m_{1}\sum_{i=1}^{n}\Big{(}\max\left\\{\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{-}},\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{+}}\right\\}+\mathcal{H}(k_{i}^{3})\max\left\\{\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{-}},\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{+}}\right\\}\Big{)}}$

where

$\displaystyle
K(t):=\left\\{(\xi_{1},...\xi_{n})\in\mathbb{R}^{N}:\displaystyle\sum_{i=1}^{n}\bigg{(}\min\left\\{|\xi_{i}|_{p_{i}^{\partial}(x)}^{{(p_{i}^{\partial})^{-}}},|\xi_{i}|_{p_{i}^{\partial}(x)}^{{(p_{i}^{\partial})^{+}}}\right\\}+\mathcal{H}(k_{i}^{3})\min\left\\{|\xi_{i}|_{q_{i}^{\partial}(x)}^{{(q_{i}^{\partial})^{-}}},|\xi_{i}|_{q_{i}^{\partial}(x)}^{{(q_{i}^{\partial})^{+}}}\right\\}\bigg{)}\leq
t\right\\},$ (2.3)

and

$m_{0}^{\star}=\displaystyle m_{0}\min_{1\leq i\leq
n}\Big{\\{}\displaystyle\frac{\min\big{\\{}\displaystyle\min\\{k_{1i}^{0},k_{2i}^{0}\\},\displaystyle\min\\{k_{1i}^{2},k_{2i}^{2}\\}\big{\\}}}{\displaystyle\max\\{\beta_{1i},\beta_{2i}\\}}\Big{\\}}\text{
and }\displaystyle s=\max\left\\{p_{i}^{+}\min_{1\leq i\leq
n}\left\\{c_{p_{i}(x)}^{{p_{i}^{-}}},c_{p_{i}(x)}^{{p_{i}^{+}}}\right\\},p_{i}^{+}\min_{1\leq
i\leq
n}\left\\{c_{q_{i}(x)}^{{q_{i}^{-}}},c_{q_{i}(x)}^{{q_{i}^{+}}}\right\\}\right\\},$
(2.4)

with $t>0$, such that $c_{p_{i}(x)}$ and $c_{q_{i}(x)}$ representing the
constants defined in Proposition 2.4.

##  3\. Main results

The following lemmas are needed in the proof of our main results, namely
Theorem 3.3 below.

######  Lemma 3.1.

The functional $\Phi$ is continuously GÃ¢teaux differentiable and sequentially
weakly lower semi-continuous, coercive and its GÃ¢teaux derivative admits a
continuous inverse on $X^{\star}$.

###### Proof.

It is well known that the functional $\Phi$ is well defined and is a
continuously GÃ¢teaux differentiable functional whose derivative at the point
$u=(u_{1},\dots,u_{n})\in X$ is the functional $\Phi^{\prime}(u)$ given by

$\displaystyle\langle\Phi^{\prime}\left(u\right),\varphi\rangle=\sum_{i=1}^{n}\langle\Phi_{i}^{\prime}\left(u_{i}\right),\varphi_{i}\rangle,$

where

$\displaystyle\langle\Phi_{i}^{\prime}(u_{i}),\varphi_{i}\rangle=M_{i}\Big{(}\mathcal{B}_{i}(u_{i})\Big{)}\int_{\mathbb{R}^{N}}\Big{(}\mathcal{A}_{1i}(\nabla
u_{i})\nabla\varphi_{i}+w_{i}(x)\mathcal{A}_{2i}(u_{i})\varphi_{i}\Big{)}dx$,

for every $\varphi=(\varphi_{1},...,\varphi_{n})\in X$ and for all
$i\in\\{1,\dots,n\\}$. Let us show that $\Phi$ is coercive. By using
$(\textbf{H}_{2})$, $(\textbf{H}_{4})$ and $(\textbf{M})$, we have for all
$u=(u_{1},\dots,u_{n})\in X$,

$\displaystyle\begin{aligned}
\Phi(u)&=\sum_{i=1}^{n}\Phi_{i}(u_{i})=\sum_{i=1}^{n}\widehat{M}_{i}\Big{(}\mathcal{B}_{i}(u_{i})\Big{)}\\\
&\geq\sum_{i=1}^{n}\frac{m_{0}}{p_{i}^{+}}\int_{\mathbb{R}^{N}}\Big{(}\mathcal{A}_{1i}(|\nabla
u_{i}|)+w_{i}(x)\mathcal{A}_{2i}(|u_{i}|)\Big{)}dx\\\
&\geq\sum_{i=1}^{n}\frac{m_{0}}{p_{i}^{+}}\int_{\mathbb{R}^{N}}\Big{[}\frac{1}{\beta_{1i}}a_{1i}(|\nabla
u_{i}|^{p_{i}(x)})|\nabla
u_{i}|^{p_{i}(x)}+\frac{1}{\beta_{2i}}w_{i}(x)a_{2i}(|u_{i}|^{p_{i}(x)})|u_{i}|^{p_{i}(x)}\Big{]}dx\\\
&\geq\sum_{i=1}^{n}\frac{m_{0}}{p_{i}^{+}\max\\{\beta_{1i},\beta_{2i}\\}}\int_{\mathbb{R}^{N}}\Big{[}a_{1i}(|\nabla
u_{i}|^{p_{i}(x)})|\nabla
u_{i}|^{p_{i}(x)}+w_{i}(x)a_{2i}(|u_{i}|^{p_{i}(x)})|u_{i}|^{p_{i}(x)}\Big{]}dx\\\
&\geq\sum_{i=1}^{n}\frac{m_{0}\min\\{\min\\{k_{1i}^{0},k_{2i}^{0}\\},\min\\{k_{1i}^{2},k_{2i}^{2}\\}\\}}{p_{i}^{+}\max\\{\beta_{1i},\beta_{2i}\\}}\cdot\Big{(}\rho_{p_{i}(x)w_{i}}(u_{i})+\mathcal{H}(k_{i}^{3})\cdot\rho_{q_{i}(x)w_{i}}(u_{i})\Big{)},\end{aligned}$

Hence, by inequalities (2.1) and $\eqref{inq2}$, we obtain

$\displaystyle\begin{aligned}
\Phi(u)&\geq\sum_{i=1}^{n}\frac{m_{0}\min\\{\min\\{k_{1i}^{0},k_{2i}^{0}\\},\min\\{k_{1i}^{2},k_{2i}^{2}\\}\\}}{p_{i}^{+}\max\\{\beta_{1i},\beta_{2i}\\}}\cdot\Big{(}\min\\{\|u_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{-}},\|u_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{+}}\\}+\mathcal{H}(k_{i}^{3})\cdot\min\\{\|u_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{-}},\|u_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{+}}\\}\Big{)}.\end{aligned}$

This shows that $\Phi(u)\longrightarrow+\infty$ as
$\|u\|\longrightarrow+\infty$ that is, $\Phi$ is coercive on $X$. Now, in
order to show that the functional $\Phi^{\prime}:X\longrightarrow X^{\star}$
is strictly monotone, it suffices to prove that $\Phi$ is strictly convex.

For all $i\in\\{1,\dots,n\\}$, the functional
$\mathcal{B}_{i}:X_{i}\longrightarrow\mathbb{R}$ defined in (1.3) is clearly a
GÃ¢teaux differentiable at any $u_{i}\in X_{i}$, and its derivative is given
by

$\displaystyle\displaystyle\langle\mathcal{B}_{i}^{\prime}(u_{i}),\varphi_{i}\rangle$
$\displaystyle=\int_{\mathbb{R}^{N}}\Big{(}\mathcal{A}_{1i}(|\nabla
u_{i}|^{p_{i}(x)})\nabla\varphi_{i}+w_{i}(x)\mathcal{A}_{2i}(|u_{i}|^{p_{i}(x)})\varphi_{i}\Big{)}dx$
$\displaystyle=\int_{\mathbb{R}^{N}}\Big{(}a_{1i}(|\nabla
u_{i}|^{p_{i}(x)})|\nabla u_{i}|^{p_{i}(x)-2}\nabla
u_{i}.\nabla\varphi_{i}+w_{i}(x)a_{2i}(|u_{i}|^{p_{i}(x)})|u_{i}|^{p_{i}(x)-2}u_{i}\varphi_{i}\Big{)}dx,$

for all $\varphi_{i}\in X_{i}$.

Taking into account the elementary inequalities (see, e.g., Auxiliary Results
in [22]) for any $\varrho,\zeta\in\mathbb{R}^{N}$

$\Big{(}a_{ji}(|\varrho|^{p_{i}(x)})|\varrho|^{p_{i}(x)-2}\varrho-
a_{ji}(|\zeta|^{p_{i}(x)})|\zeta|^{p_{i}(x)-2}\zeta\Big{)}\cdot\left(\varrho-\zeta\right)\geq\begin{cases}C_{p_{i}}|\varrho-\zeta|^{p_{i}(x)}&\text{if
}p_{i}(x)\geq 2\\\\[4.0pt]
C_{p_{i}}\frac{|\varrho-\zeta|^{2}}{(|\varrho|+|\zeta|)^{p_{i}(x)-2}},\;(\varrho,\zeta)\neq(0,0)&\text{if
}1<p_{i}(x)<2,\end{cases}$ (3.1)

where $\cdot$ denotes the standard inner product in $\mathbb{R}^{N}$.
Therefore, we have

$\displaystyle\langle\mathcal{B}_{i}^{\prime}(u_{i})-\mathcal{B}_{i}^{\prime}(v_{i}),u_{i}-v_{i}\rangle>0$,

for all $u_{i}\neq v_{i}\in X_{i}$, which means that
$\mathcal{B}_{i}^{\prime}$ is strictly monotone. So, by [33, Proposition
25.10], $\mathcal{B}_{i}$ is strictly convex. Moreover, since the Kirchhoff
function $M_{i}$ is nondecreasing, $\widehat{M}_{i}$ is convex in
$[0,+\infty[$. Thus, for every $u_{i},v_{i}\in X_{i}$ with $u_{i}\neq v_{i}$,
and every $s,t\in(0,1)$ with $s+t=1$, we have

$\widehat{M}_{i}(\mathcal{B}_{i}(su_{i}+tv_{i}))<\widehat{M}_{i}(s\mathcal{B}_{i}(u_{i})+t\mathcal{B}_{i}(v_{i}))\leq
s\widehat{M}_{i}(\mathcal{B}_{i}(u_{i}))+t\widehat{M}_{i}(\mathcal{B}_{i}(v_{i})).$

This shows that $\Phi_{i}$ is strictly convex in
$W_{w_{i}}^{1,p_{i}(x)}\left(\mathbb{R}^{N}\right)\cap
W^{1,\gamma_{i}(x)}_{w_{i}}\left(\mathbb{R}^{N}\right)$ for all
$i\in\\{1,\dots,n\\}$. Hence, $\Phi$ is strictly convex in $X$, and therefore
$\Phi^{\prime}=\displaystyle\sum_{i=1}^{n}\Phi_{i}^{\prime}$ is strictly
monotone.

It’s clear that $\Phi^{\prime}$ is an injection since $\Phi^{\prime}$ is
strictly monotone operator in $X$.
Moreover, since we have

$\displaystyle\begin{aligned}
\displaystyle\lim_{\|u\|\longrightarrow+\infty}\frac{\langle\Phi^{\prime}(u),u\rangle}{\|u\|}&=\displaystyle\lim_{\|(u_{1},...,u_{n})\|\to+\infty}\frac{\displaystyle\sum_{i=1}^{n}M_{i}\left(\mathcal{B}_{i}(u_{i})\right)\int_{\mathbb{R}^{N}}\Big{(}\mathcal{A}_{1i}(\nabla
u_{i})\nabla
u_{i}+w_{i}(x)\mathcal{A}_{2i}(u_{i})u_{i}\Big{)}\,dx}{\|(u_{1},...,u_{n})\|}\\\
&\geq\displaystyle\lim_{\|(u_{1},...,u_{n})\|\to+\infty}\frac{m_{0}\displaystyle\sum_{i=1}^{n}\int_{\mathbb{R}^{N}}\Big{(}a_{1i}(|\nabla
u_{i}|^{p_{i}(x)})|\nabla
u_{i}|^{p_{i}(x)}+w_{i}(x)a_{2i}(|u_{i}|^{p_{i}(x)})|u_{i}|^{p_{i}(x)}\Big{)}\,dx}{\|(u_{1},...,u_{n})\|}\\\
&\geq\displaystyle\lim_{\|(u_{1},...,u_{n})\|\to+\infty}\frac{m_{0}\displaystyle\min_{1\leq
i\leq
n}\Big{\\{}\min\\{k_{1i}^{0},k_{2i}^{0}\\},\min\\{k_{1i}^{2},k_{2i}^{2}\\}\Big{\\}}\sum_{i=1}^{n}\Big{(}\|u_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{+}}+\mathcal{H}(k_{i}^{3})\cdot\|u_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{+}}\Big{)}}{\displaystyle\max_{1\leq
i\leq
n}\left\\{\|u_{i}\|_{p_{i}(x),w_{i}}+\mathcal{H}(\kappa_{i}^{3})\|u_{i}\|_{q_{i}(x),w_{i}}\right\\}}=+\infty.\end{aligned}$

Then, we deduce that $\Phi^{\prime}$ is coercive. Thus $\Phi^{\prime}$ is a
surjection. Now, since $\Phi^{\prime}$ is semicontinuous in $X$, then by
applying Minty-Browder theorem (Theorem 26.A of [33]), we conclude that
$\Phi^{\prime}$ admits a continuous inverse on $X^{\star}$. Moreover, the
monotonicity of $\Phi^{\prime}$ on $X^{\star}$ ensures that $\Phi$ is
sequentially lower semi-continuously on $X$ (see [33], Proposition 25. 20).
The proof of the lemma is complete. ∎

######  Lemma 3.2.

Under assumptions $(\mathcal{F}_{1})$ and $(\mathcal{F}_{2})$, the functional
$\Psi$ is well defined and is of class $C^{1}$ on $X$. Moreover, its
derivative is given by

$\displaystyle\displaystyle\langle\Psi^{\prime}(u),\varphi\displaystyle\rangle=\sum_{i=1}^{n}\frac{\partial
F}{\partial u_{i}}(x,u)\varphi_{i}$, $\forall
u=(u_{1},\dots,u_{n}),\varphi=(\varphi_{1},\dots,\varphi_{n})\in X$.

Furthermore, $\Psi^{\prime}$ is compact from $X$ to $X^{\star}$.

###### Proof.

We start by showing that the functional $F$ is well defined on $X$. Indeed,
for all $u=(u_{1},\dots,u_{n})\in X$, we have in virtue of $(\mathcal{F}_{1})$
and $(\mathcal{F}_{2})$,

$\displaystyle\begin{aligned}
F(x,u)&=\sum_{i=1}^{n}\int_{0}^{u_{i}}\frac{\partial F}{\partial
s}(x,u_{1},\dots,s,\dots,u_{n})ds+F(x,0,\dots,0)dx\\\ &\leq
c_{1}\left[\sum_{i=1}^{n}\left(\sum_{j=1}^{n}b_{ij}(x)\left|u_{j}(x)\right|^{\mu_{ij}-1}\left|u_{i}(x)\right|\right)\right],\end{aligned}$

Then,

$\displaystyle\int_{\mathbb{R}^{N}}F\left(x,u_{1},\ldots,u_{n}\right)dx\leq
c_{2}\left[\sum_{i=1}^{n}\left(\int_{\mathbb{R}^{N}}\sum_{j=1}^{n}b_{ij}(x)\left|u_{j}(x)\right|^{\mu_{ij}-1}\left|u_{i}(x)\right|dx\right)\right].$

If we consider the fact that
$W^{1,\gamma(x)}\left(\mathbb{R}^{N}\right)\hookrightarrow
L^{\mu(x)}\left(\mathbb{R}^{N}\right)$, for $\mu(x)>1$, then there exists
$c>0$ such that

$\displaystyle||u|^{\mu}|_{\gamma(x)}=|u|_{\mu\gamma(x)}^{\mu}\leq
c\|u\|_{\gamma(x)}^{\mu}$,

and if we apply Propositions 2.1,2.2 and 2.5 and take $b_{ii}\in
L^{\alpha_{i}(x)},b_{ij}\in L^{\alpha_{ij}(x)}$ if $i\neq j$, then we have

$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}F(x,u_{1},\ldots,u_{n})$
$\displaystyle\leq
c_{3}\Big{[}\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}|b_{ij}|_{\alpha_{ij}(x)}||u_{j}|^{\mu_{ij}-1}|_{\gamma_{j}^{\partial}(x)}|u_{i}|_{\gamma_{i}^{\partial}(x)}\Big{)}\Big{]}\quad\quad(3.1)$
(3.2) $\displaystyle\leq
c_{3}\Big{[}\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}|b_{ij}|_{\alpha_{ij}(x)}|u_{j}|_{(\mu_{ij}-1)\gamma_{j}^{\partial}(x)}^{\mu_{ij}-1}|u_{i}|_{\gamma_{i}^{\partial}(x)}\Big{)}\Big{]}\quad\quad(3.2)$
(3.3) $\displaystyle\leq
c_{3}\Big{[}\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}|b_{ij}|_{\alpha_{ij}(x)}\|u_{j}\|_{\gamma_{j}(x)}^{\mu_{ij}-1}\|u_{i}\|_{\gamma_{i}(x)}\Big{)}\Big{]}<\infty.\quad\quad(3.3)$
(3.4)

.

Hence, $\Psi$ is well defined. Moreover, one can easily see that
$\Psi^{\prime}$ is also well defined on $X$. Indeed, using $(\mathcal{F}_{2})$
for all $\varphi=\left(\varphi_{1},\dots,\varphi_{n}\right)\in X$, we have

$\displaystyle\begin{aligned}
\Psi^{\prime}(u)\varphi&=\sum_{i=1}^{n}\int_{\mathbb{R}^{N}}\frac{\partial
F}{\partial u_{i}}\left(x,u_{1},\ldots,u_{n}\right)\varphi_{i}dx\\\
&\leq\sum_{i=1}^{n}\left(\sum_{j=1}^{n}b_{ij}(x)\left|u_{j}(x)\right|^{\mu_{ij}-1}\right)\left|\varphi_{i}(x)\right|dx.\end{aligned}$

Following HÃ¶lder inequality, we obtain

$\displaystyle\Psi^{\prime}(u)\varphi\leq
c_{4}\Big{[}\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}|b_{ij}|_{\alpha_{ij}(x)}||u_{j}|^{\mu_{ij}-1}|_{\gamma_{j}^{\partial}(x)}|\varphi_{i}|_{\gamma_{i}^{\partial}(x)}\Big{)}\Big{]}$.

The above propositions yield

$\displaystyle\Psi^{\prime}(u)\varphi\leq
c\left[\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\left|b_{ij}\right|_{\beta_{ij}(x)}\left\|u_{j}\right\|_{\gamma_{j}(x)}^{\mu_{ij}-1}\left\|\varphi_{i}\right\|_{\gamma_{i}(x)}\right)\right]<\infty$.

Now let us show that $\Psi$ is differentiable in the sense of Frechet, that
is, for fixed $u=\left(u_{1},\ldots,u_{n}\right)\in X$ and given
$\varepsilon>0$, there must be a
$\delta=\delta_{\varepsilon,u_{1},\ldots,u_{n}}>0$ such that

$\displaystyle\left|\Psi\left(u_{1}+\varphi_{1},\ldots,u_{n}+\varphi_{n}\right)-\Psi\left(u_{1},\ldots,u_{n}\right)-\Psi^{\prime}\left(u_{1},\ldots,u_{n}\right)\left(\varphi_{1},\ldots,\varphi_{n}\right)\right|\leq\varepsilon\sum_{i=1}^{n}\left(\left\|\varphi_{i}\right\|_{\gamma_{i}(x)}\right),$

for all $\varphi=\left(\varphi_{1},\ldots,\varphi_{n}\right)\in X$ with
$\displaystyle\sum_{i=1}^{n}\left(\left\|\varphi_{i}\right\|_{\gamma_{i}(x)}\right)\leq\delta$.

Let $B_{R}=\Big{\\{}x\in\mathbb{R}^{N}:|x|<R\Big{\\}}$ be the ball of radius
$R$ which is centered at the origin of $\mathbb{R}^{N}$ and denote
$B_{R}^{\prime}=\mathbb{R}^{N}\backslash B_{R}$. Moreover, let us define the
functional
$\Psi_{R}:\displaystyle\prod_{i=1}^{n}\Big{(}W_{w_{i}}^{1,p_{i}(x)}\left(B_{R}\right)\cap
W_{w_{i}}^{1,\gamma_{i}(x)}\left(B_{R}\right)\Big{)}\longrightarrow\mathbb{R}$
as follows:

$\displaystyle\Psi_{R}(u)=\int_{B_{R}}F\left(x,u_{1}(x),\ldots,u_{n}(x)\right)dx,\quad\forall
u=(u_{1},\dots,u_{n})$.

If we consider $(\mathcal{F}_{1})$ and $(\mathcal{F}_{2})$, it is easy to see
that $\displaystyle\Psi_{R}\in
C^{1}\Big{(}\prod_{i=1}^{n}W_{w_{i}}^{1,p_{i}(x)}\left(B_{R}\right)\cap
W_{w_{i}}^{1,\gamma_{i}(x)}\left(B_{R}\right)\Big{)}$, and in addition for all
$\displaystyle\varphi=\left(\varphi_{1},\ldots,\varphi_{n}\right)\in\prod_{i=1}^{n}\Big{(}W_{w_{i}}^{1,p_{i}(x)}\left(B_{R}\right)\cap
W_{w_{i}}^{1,\gamma_{i}(x)}\left(B_{R}\right)\Big{)}$, we have

$\displaystyle\Psi_{R}^{\prime}(u)\varphi=\sum_{i=1}^{n}\int_{B_{R}}\frac{\partial
F}{\partial u_{i}}\left(x,u_{1}(x),\ldots,u_{n}(x)\right)\varphi_{i}(x)dx$.

Also as we know, the operator
$\displaystyle\Psi_{R}^{\prime}:\prod_{i=1}^{n}\Big{(}W_{w_{i}}^{1,p_{i}(x)}\left(B_{R}\right)\cap
W_{w_{i}}^{1,\gamma_{i}(x)}\left(B_{R}\right)\Big{)}\longrightarrow\Big{(}\prod_{i=1}^{n}\Big{(}W_{w_{i}}^{1,p_{i}(x)}\left(B_{R}\right)\cap
W_{w_{i}}^{1,\gamma_{i}(x)}\left(B_{R}\right)\Big{)}\Big{)}^{\star}$ is
compact [20]. Then, for all
$u=\left(u_{1},\ldots,u_{n}\right),\varphi=\left(\varphi_{1},\ldots,\varphi_{n}\right)\in
X$, we can write

$\displaystyle\begin{aligned}
&\left|\Psi\left(u_{1}+\varphi_{1},\ldots,u_{n}+\varphi_{n}\right)-\Psi\left(u_{1},\ldots,u_{n}\right)-\Psi^{\prime}\left(u_{1},\ldots,u_{n}\right)\left(\varphi_{1},\ldots,\varphi_{n}\right)\right|\\\
&\leq\left|\Psi_{R}\left(u_{1}+\varphi_{1},\ldots,u_{n}+\varphi_{n}\right)-\Psi_{R}\left(u_{1},\ldots,u_{n}\right)-\Psi_{R}^{\prime}\left(u_{1},\ldots,u_{n}\right)\left(\varphi_{1},\ldots,\varphi_{n}\right)\right|\\\
&\quad+\Big{|}\int_{B_{R}^{\prime}}\left(F\left(x,u_{1}+\varphi_{1},\ldots,u_{n}+\varphi_{n}\right)-F\left(x,u_{1},\ldots,u_{n}\right)-\right.\left.\sum_{i=1}^{n}\int_{B_{R}}\frac{\partial
F}{\partial
u_{i}}\left(x,u_{1},\ldots,u_{n}\right)\varphi_{i}\right)dx\Big{|}.\end{aligned}$

By virtue of mean-value theorem, there exist
$\left.\xi_{1},\ldots,\xi_{n}\in\right]0,1[$ such that

$\displaystyle\begin{gathered}\left|\int_{B_{R}^{\prime}}\left(F\left(x,u_{1}+\varphi_{1},\ldots,u_{n}+\varphi_{n}\right)-F\left(x,u_{1},\ldots,u_{n}\right)\right)-\sum_{i=1}^{n}\int_{B_{R}}\frac{\partial
F}{\partial u_{i}}\left(x,u_{1},\ldots,u_{n}\right)\varphi_{i}dx\right|\\\
=\left|\int_{B_{R}^{\prime}}\left(\sum_{i=1}^{n}\frac{\partial F}{\partial
u_{i}}\left(x,u_{1},\ldots,u_{i}+\xi_{i}\varphi_{i},\ldots,u_{n}\right)\varphi_{i}-\sum_{i=1}^{n}\frac{\partial
F}{\partial
u_{i}}\left(x,u_{1},\ldots,u_{n}\right)\right)dx\right|.\end{gathered}$

Using the condition $(\mathcal{F}_{2})$, we have

$\displaystyle\left|\int_{B_{R}^{\prime}}\left(F\left(x,u_{1}+\varphi_{1},\ldots,u_{n}+\varphi_{n}\right)-F\left(x,u_{1},\ldots,u_{n}\right)\right)-\sum_{i=1}^{n}\int_{B_{R}}\frac{\partial
F}{\partial
u_{i}}\left(x,u_{1},\ldots,u_{n}\right)\varphi_{i}dx\right|\leq\left|\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\int_{B_{R}^{\prime}}b_{ij}(x)\left(\left|u_{j}+\xi_{j}\varphi_{j}\right|^{\mu_{ij}-1}-\left|u_{j}\right|^{\mu_{ij}-1}\right)\varphi_{i}dx\right)\right|.$

Using the elementary inequality $|a+b|^{s}\leq
2^{s-1}\left(|a|^{s}+|b|^{s}\right)$ for $a,b\in\mathbb{R}^{N}$, we can write

$\displaystyle\begin{gathered}\leq\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\left(\left(2^{\mu_{ij}-1}-1\right)\int_{B_{R}^{\prime}}b_{ij}(x)\left|u_{j}\right|^{\mu_{ij}-1}\left|\varphi_{i}\right|dx\right.\left.+\left(\xi_{j}2\right)^{\mu_{ij}-1}\int_{B_{R}^{\prime}}b_{ij}(x)\left|\varphi_{j}\right|^{\mu_{ij}-1}\left|\varphi_{i}\right|dx\right)\right)\end{gathered}$.

Then, applying Propositions 2.2, 2.3 and 2.5, we have

$\displaystyle\leq\sum_{i=1}^{n}c\left(\sum_{j=1}^{n}\left(\left|b_{ij}(x)\right|_{\alpha_{ij}}\left\|u_{j}\right\|_{\gamma_{1}^{\partial}(x)}^{\mu_{ij}-1}+\left|b_{ij}(x)\right|_{\alpha_{ij}}\left\|\varphi_{j}\right\|_{\gamma_{j}^{\partial}(x)}^{\mu_{ij}-1}\right)\right)\left\|\varphi_{i}\right\|_{\gamma_{i}(x)}$

and by the fact that

$\displaystyle\left|b_{ii}(x)\right|_{L^{\alpha_{i}}\left(B_{R}^{\prime}\right)}\longrightarrow
0$,

and

$\displaystyle\left|b_{ij}(x)\right|_{L^{\alpha_{i}}\left(B_{R}^{\prime}\right)}\longrightarrow
0$,

for all $1\leq i,j\leq n$, as $R\rightarrow\infty$, and for $R$ sufficiently
large, we obtain the estimate

$\displaystyle\Big{|}\int_{B_{R}^{\prime}}\left(F\left(x,u_{1}+\varphi_{1},\ldots,u_{n}+\varphi_{n}\right)-F\left(x,u_{1},\ldots,u_{n}\right)-\sum_{i=1}^{n}\frac{\partial
F}{\partial
u_{i}}\left(x,u_{1},\ldots,u_{n}\right)\varphi_{i}\right)dx\Big{|}\leq\varepsilon\sum_{i=1}^{n}\left(\left\|\varphi_{i}\right\|_{\gamma_{i}(x)}\right)$.

It only remains to show that $\Psi^{\prime}$ is continuous on $X$. Let
$u_{m}=\left(u_{m,1},\ldots,u_{m,n}\right)$ be such that $u_{m}\rightarrow u$
as $m\rightarrow\infty$. Then, for
$\varphi=\left(\varphi_{1},\ldots,\varphi_{n}\right)\in X$, we have

$\displaystyle\begin{aligned}
\left|\Psi^{\prime}\left(u_{m}\right)\varphi-\Psi^{\prime}(u)\varphi\right|&\leq\left|\Psi_{R}^{\prime}\left(u_{m}\right)\varphi-\Psi_{R}^{\prime}(u)\varphi\right|\\\
&+\sum_{i=1}^{n}\int_{B_{R}^{\prime}}\left|\left(\frac{\partial F}{\partial
u_{i}}\left(x,u_{m,1},\ldots,u_{m,n}\right)\varphi_{i}-\frac{\partial
F}{\partial
u_{i}}\left(x,u_{1},\ldots,u_{n}\right)\varphi_{i}\right)dx\right|\end{aligned}$

Since $\Psi_{R}^{\prime}$ is continuous on
$\displaystyle\prod_{i=1}^{n}\Big{(}W_{w_{i}}^{1,p_{i}(x)}\left(B_{R}\right)\cap
W_{w_{i}}^{1,\gamma_{i}(x)}\left(B_{R}\right)\Big{)}$ (see [20]), we have

$\displaystyle\left|\Psi_{R}^{\prime}\left(u_{m}\right)\varphi-\Psi_{R}^{\prime}(u)\varphi\right|\longrightarrow
0\quad\text{ as }m\rightarrow\infty$.

Now, using $(\mathcal{F}_{2})$ once again and taking into account that the
other terms on the right-hand side of the above inequality tend to zero, we
conclude that $\Psi^{\prime}$ is continuous on $X$.

As for the compactness of $\Psi^{\prime}$, let
$u_{m}=\left(u_{m,1},\ldots,u_{m,n}\right)$ be a bounded sequence in $X$.
Then, there exists a subsequence (we also denote it as
$u_{m}=\left(u_{m,1},\ldots,u_{m,n}\right)$) which converges weakly in $X$ to
$u=\left(u_{1},\ldots,u_{n}\right)\in X$. Then, if we use the same arguments
as above, we have

$\displaystyle\begin{aligned}
\left|\Psi^{\prime}\left(u_{m}\right)\varphi-\Psi^{\prime}(u)\varphi\right|&\leq\left|\Psi_{R}^{\prime}\left(u_{m}\right)\varphi-\Psi_{R}^{\prime}(u)\varphi\right|\\\
&+\sum_{i=1}^{n}\int_{B_{R}^{\prime}}\left|\left(\frac{\partial F}{\partial
u_{i}}\left(x,u_{m,1},\ldots,u_{m,n}\right)\varphi_{i}-\frac{\partial
F}{\partial
u_{i}}\left(x,u_{1},\ldots,u_{n}\right)\varphi_{i}\right)dx\right|.\end{aligned}$

Since the restriction operator is continuous, we have $u_{m}\rightharpoonup u$
in
$\displaystyle\prod_{i=1}^{n}\Big{(}W_{w_{i}}^{1,p_{i}(x)}\left(B_{R}\right)\cap
W_{w_{i}}^{1,\gamma_{i}(x)}\left(B_{R}\right)\Big{)}$. Because of the
compactness of $\Psi^{\prime}$, the first expression on the right-hand side of
the inequality tends to 0 , as $m\longrightarrow\infty$, and, as above, for
sufficiently large $R$ we obtain

$\displaystyle\sum_{i=1}^{n}\int_{B_{R}^{\prime}}\left|\left(\frac{\partial
F}{\partial
u_{i}}\left(x,u_{m,1},\ldots,u_{m,n}\right)\varphi_{i}-\frac{\partial
F}{\partial
u_{i}}\left(x,u_{1},\ldots,u_{n}\right)\varphi_{i}\right)\right|dx\longrightarrow
0.$

This implies $\Psi^{\prime}$ is compact from $X$ to $X^{\star}$. ∎

######  Theorem 3.3.

Under assumptions $(\mathcal{F}_{1})-(\mathcal{F}_{3})$, the system (1.1)
admits at least three distinct weak solutions in $X$ for each $\lambda$ that
belongs to the interval

$\displaystyle\Lambda_{r}=\Bigg{]}\frac{\displaystyle
m_{1}\sum_{i=1}^{n}\max\left\\{\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{-}},\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{+}}\right\\}+\mathcal{H}(k_{i}^{3})\max\left\\{\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{-}},\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{+}}\right\\}}{\displaystyle\int_{\mathbb{R}^{N}}F(x,z_{1},\dots,z_{n})dx},\frac{r}{\displaystyle\int_{\mathbb{R}^{N}}\sup_{(\xi_{1},\dots,\xi_{n})\in
K(\frac{sr}{m_{0}^{\star}})}F(x,\xi_{1},\dots,\xi_{n})dx}\Bigg{[}.$

###### Proof.

By Lemma 3.1, $\Phi$ is coercive, and by definitions of $\Phi$ and $\Psi$ and
from hypothesis $(\mathcal{F}_{1})$, we have
$\Phi(0,\dots,0)=\Psi(0,\dots,0)=0$. Moreover, the required hypothesis
$\Phi(\bar{x})>r$ follows from $(C_{1})$ and the definition of $\Phi$ by
choosing $\bar{x}=(z_{1},\dots,z_{n})$. On the contrary, by applying
Proposition 2.4 for $u=(u_{1},\dots,u_{n})\in X$, we have

$\displaystyle\frac{1}{s}\Bigg{(}\sum_{i=1}^{n}\Big{(}\min\Big{\\{}|u_{i}|_{p_{i}^{\partial}}^{p_{i}^{-}},|u_{i}|_{p_{i}^{\partial}}^{p_{i}^{+}}\Big{\\}}+\mathcal{H}(\kappa_{i}^{3})\cdot\min\Big{\\{}|u_{i}|_{q_{i}^{\partial}}^{q_{i}^{-}},|u_{i}|_{q_{i}^{\partial}}^{q_{i}^{+}}\Big{\\}}\Big{)}\Bigg{)}\\\
\leq\sum_{i=1}^{n}\frac{1}{p_{i}^{+}}\Big{(}\min\Big{\\{}\|u_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{-}},\|u_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{+}}\Big{\\}}+\mathcal{H}(k_{i}^{3})\cdot\min\Big{\\{}\|u_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{-}},\|u_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{+}}\Big{\\}}\Big{)}.$
(3.5)

with $\displaystyle s=\max_{1\leq i\leq
n}\Big{\\{}p_{i}^{+}\displaystyle\min\\{c_{p_{i}(x)}^{p_{i}^{-}},c_{p_{i}(x)}^{p_{i}^{+}}\\},p_{i}^{+}\displaystyle\min\\{c_{q_{i}(x)}^{q_{i}^{-}},c_{q_{i}(x)}^{q_{i}^{+}}\\}\Big{\\}}$,
defined in $\eqref{defs}$.

Now, from (3.5), we obtain for $r>0$

$\displaystyle\begin{aligned}
\Phi^{-1}(]-\infty,r])&=\Big{\\{}u=(u_{1},\dots,u_{n})\in
X:\Phi(u_{1},\dots,u_{n})\leq r\Big{\\}}\\\
&\subseteq\Big{\\{}u=(u_{1},\dots,u_{n})\in
X:\displaystyle\sum_{i=1}^{n}\frac{m_{i}^{0}\min\\{\displaystyle\min\\{k_{1i}^{0},k_{2i}^{0}\\},\displaystyle\min\\{k_{1i}^{2},k_{2i}^{2}\\}\\}}{p_{i}^{+}\displaystyle\max\\{\beta_{1i},\beta_{2i}\\}}\\\
&\quad\cdot\Big{(}\min\\{\|u_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{-}},\|u_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{+}}\\}+\mathcal{H}(k_{i}^{3})\cdot\min\\{\|u_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{-}},\|u_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{+}}\\}\Big{)}\leq
r\Big{\\}}\\\ &\subseteq\displaystyle\Big{\\{}\frac{\displaystyle
m_{0}\min_{1\leq i\leq n}\\{\kappa_{i}\\}}{\displaystyle\max_{1\leq i\leq
n}\Big{\\{}p_{i}^{+}\displaystyle\min\\{c_{p_{i}(x)}^{p_{i}^{-}},c_{p_{i}(x)}^{p_{i}^{+}}\\},p_{i}^{+}\displaystyle\min\\{c_{q_{i}(x)}^{q_{i}^{-}},c_{q_{i}(x)}^{q_{i}^{+}}\\}\Big{\\}}}\sum_{i=1}^{n}\Big{(}\min\\{|\xi_{i}|_{p_{i}^{\partial}(x)}^{{p_{i}^{-}}},|\xi_{i}|_{p_{i}^{\partial}(x)}^{{p_{i}^{+}}}\\}\\\
&+\mathcal{H}(k_{i}^{3})\min\\{|\xi_{i}|_{q_{i}^{\partial}(x)}^{{q_{i}^{-}}},|\xi_{i}|_{q_{i}^{\partial}(x)}^{{q_{i}^{+}}}\\}\Big{)}\leq
r\Big{\\}}\\\ &=\displaystyle
K\Big{(}\frac{sr}{m_{0}^{\star}}\Big{)}\end{aligned}$

where $K(t)$ is defined in $\eqref{defK}$, and $\kappa_{i}$ and
$m_{0}^{\star}$ are defined as follows

$\displaystyle\begin{aligned}
\kappa_{i}&=\displaystyle\frac{\min\\{\displaystyle\min\\{k_{1i}^{0},k_{2i}^{0}\\},\displaystyle\min\\{k_{1i}^{2},k_{2i}^{2}\\}\\}}{\displaystyle\max\\{\beta_{1i},\beta_{2i}\\}},\\\
m_{0}^{\star}&=\displaystyle m_{0}\min_{1\leq i\leq
n}\\{\kappa_{i}\\}.\end{aligned}$

Then, we get

$\displaystyle\begin{aligned}
\sup_{(u_{1},\dots,u_{n})\in\Phi^{-1}(]-\infty,r])}\Psi(u_{1},\dots,u_{n})&=\sup_{(\xi_{1},\dots,\xi_{n})\in
K(\frac{sr}{m_{0}^{\star}})}\int_{\mathbb{R}^{N}}F(x,u_{1},\dots,u_{n})dx\\\
&\leq\int_{\mathbb{R}^{N}}\sup_{(\xi_{1},\dots,\xi_{n})\in
K(\frac{sr}{m_{0}^{\star}})}F(x,\xi_{1},\dots,\xi_{n})dx.\end{aligned}$

Therefore from $(C_{2})$, we have

$\displaystyle\begin{aligned}
\sup_{(u_{1},\dots,u_{n})\in\Phi^{-1}(]-\infty,r])}\Psi(u_{1},\dots,u_{n})&\leq
r\frac{\displaystyle\int_{\mathbb{R}^{N}}F(x,z_{1},\dots,z_{n})dx}{\displaystyle
m_{1}\sum_{i=1}^{n}\Big{(}\max\left\\{\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{-}},\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{+}}\right\\}+\mathcal{H}(k_{i}^{3})\max\left\\{\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{-}},\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{+}}\right\\}\Big{)}}\\\
&\leq r\frac{\Psi(z_{1},\dots,z_{n})}{\Phi(z_{1},\dots,z_{n})},\end{aligned}$

from which condition $(a_{1})$ of Lemma 1.1 follows.

In the next step, we shall prove that for each $\lambda>0$, the energy
functional $E_{\lambda}=\Phi-\lambda\Psi$ is coercive. For all
$u=(u_{1},\dots,u_{n})\in X$, under the assumptions $(\textbf{H}_{2})$ and
$(\textbf{H}_{3})$ and by using inequality (3.4), we have

$\displaystyle\begin{aligned}
E_{\lambda}(u)&=\sum_{i=1}^{n}\widehat{M}_{i}\Big{(}\mathcal{B}_{i}(u_{i})\Big{)}-\lambda\int_{\mathbb{R}^{N}}F(x,u_{1},\dots,u_{n})dx\\\
&\geq\sum_{i=1}^{n}\frac{m_{0}}{p_{i}^{+}}\int_{\mathbb{R}^{N}}\Big{(}\mathcal{A}_{1i}(|\nabla
u_{i}|^{p_{i}(x)})+w_{i}(x)\mathcal{A}_{2i}(|u_{i}|^{p_{i}(x)})\Big{)}dx\\\
&\quad-\lambda\int_{\mathbb{R}^{N}}\Big{(}\sum_{i=1}^{n}\int_{0}^{u_{i}}\frac{\partial
F}{\partial s}(x,u_{1},\dots,s,\dots,u_{n})ds+F(x,0,\dots,0)dx\Big{)}\\\
&\geq\sum_{i=1}^{n}\frac{m_{0}}{p_{i}^{+}}\int_{\mathbb{R}^{N}}\Big{(}\frac{1}{\beta_{1i}}a_{1i}\big{(}|\nabla
u_{i}|^{p_{i}(x)}\big{)}|\nabla
u_{i}|^{p_{i}(x)}+\frac{1}{\beta_{2i}}w_{i}(x)a_{2i}\big{(}|u_{i}|^{p_{i}(x)}\big{)}|u_{i}|^{p_{i}(x)}\Big{)}dx\\\
&\quad-\lambda
c_{3}\Big{[}\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}|b_{ij}|_{\alpha_{ij}(x)}\|u_{j}\|_{\gamma_{j}(x)}^{\mu_{ij}-1}\|u_{i}\|_{\gamma_{i}(x)}\Big{)}\Big{]}\\\
&\geq\sum_{i=1}^{n}\frac{m_{0}}{p_{i}^{+}\max\\{\beta_{1i},\beta_{2i}\\}}\int_{\mathbb{R}^{N}}\Big{(}a_{1i}\big{(}|\nabla
u_{i}|^{p_{i}(x)}\big{)}|\nabla
u_{i}|^{p_{i}(x)}+w_{i}(x)a_{2i}\big{(}|u_{i}|^{p_{i}(x)}\big{)}|u_{i}|^{p_{i}(x)}\Big{)}dx\\\
&\quad-\lambda
c_{3}\Big{[}\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}|b_{ij}|_{\alpha_{ij}(x)}\|u_{j}\|_{\gamma_{j}(x)}^{\mu_{ij}-1}\|u_{i}\|_{\gamma_{i}(x)}\Big{)}\Big{]}\\\
&\geq\sum_{i=1}^{n}\frac{m_{0}\min\big{\\{}\min\\{k_{1i}^{0},k_{2i}^{0}\\},\min\\{k_{1i}^{2},k_{2i}^{2}\\}\big{\\}}}{p_{i}^{+}\max\\{\beta_{1i},\beta_{2i}\\}}\int_{\mathbb{R}^{N}}\Big{[}\Big{(}|\nabla
u_{i}|^{p_{i}(x)}+w_{i}(x)|u_{i}|^{p_{i}(x)}\Big{)}\\\
&\quad+\mathcal{H}(k_{i}^{3})\Big{(}|\nabla
u_{i}|^{q_{i}(x)}+w_{i}(x)|u_{i}|^{q_{i}(x)}\Big{)}\Big{]}dx-\lambda
c_{3}\Big{[}\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}|b_{ij}|_{\alpha_{ij}(x)}\|u_{j}\|_{\gamma_{j}(x)}^{\mu_{ij}-1}\|u_{i}\|_{\gamma_{i}(x)}\Big{)}\Big{]}\\\
&\geq\sum_{i=1}^{n}\frac{m_{0}\kappa_{i}}{p_{i}^{+}}\cdot\Big{(}\rho_{p_{i}(x)w_{i}}(u_{i})+\rho_{q_{i}(x)w_{i}}(u_{i})\Big{)}-\lambda
c_{3}\Big{[}\sum_{i=1}^{n}\Big{(}\sum_{j=1}^{n}|b_{ij}|_{\alpha_{ij}(x)}\|u_{j}\|_{\gamma_{j}(x)}^{\mu_{ij}-1}\|u_{i}\|_{\gamma_{i}(x)}\Big{)}\Big{]}\end{aligned}$

By using Young’s inequality and (3.1), we obtain

$\displaystyle\begin{aligned}
E_{\lambda}(u)&\geq\sum_{i=1}^{n}\frac{m_{0}\kappa_{i}}{p_{i}^{+}}\Big{(}\|u\|_{p_{i}(x),w_{i}}^{p_{i}^{-}}+\mathcal{H}(k_{i}^{3})\|u\|_{q_{i}(x),w_{i}}^{q_{i}^{-}}\Big{)}-\lambda
c_{3}\left[\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\left|b_{ij}\right|_{\alpha_{ij}(x)}\left(\frac{\mu_{ij}-1}{\mu_{ij}}\left\|u_{j}\right\|_{\gamma_{j}(x)}^{\mu_{ij}}\right.\right.\right.\left.\left.\left.+\frac{1}{\mu_{ij}}\left\|u_{i}\right\|_{\gamma_{i}(x)}^{\mu_{ij}}\right)\right)\right]\\\
&\geq\sum_{i=1}^{n}\left[\frac{m_{0}\kappa_{i}}{p_{i}^{+}}\Big{(}\|u\|_{p_{i}(x),w_{i}}^{p_{i}^{-}}+\mathcal{H}(k_{i}^{3})\|u\|_{q_{i}(x),w_{i}}^{q_{i}^{-}}\Big{)}-\lambda
c_{4}\left(\sum_{j=1}^{n}\left(\left|b_{ij}\right|_{\alpha_{ij}(x)}\left\|u_{j}\right\|_{\gamma_{j}(x)}^{\mu_{ij}}\right.\right.\right.\left.\left.\left.+\left|b_{ij}\right|_{\alpha_{ij}(x)}\left\|u_{i}\right\|_{\gamma_{i}(x)}^{\mu_{ij}}\right)\right)\right].\end{aligned}$

This shows that $(\Phi-\lambda\Psi)(u)\longrightarrow+\infty$ as
$\|u\|\longrightarrow+\infty$, since we have $\displaystyle
1<\mu_{ij}<\inf_{x\in\mathbb{R}^{N}}\gamma_{i}(x)$. Therefore, it is confirmed
that $\Phi-\lambda\Psi$ is coercive on $X$, for every parameter $\lambda>0$ in
particular for every
$\displaystyle\lambda\in\Lambda_{r}=]\frac{\Phi(\bar{x})}{\Psi(\bar{x})},\frac{r}{\displaystyle\sup_{\Phi(\bar{x})\leq
r}\Psi(\bar{x})}[$, and then the hypothesis $(a_{2})$ of Lemma 1.1 is also
proved. Now, all the hypotheses of Lemma 1.1 are satisfied. Note that the
solutions of the equation $\Phi^{\prime}(u)-\lambda\Psi^{\prime}(u)$ are
exactly the weak solutions of (1.1). Thus, for each

$\displaystyle\lambda\in\Bigg{]}\frac{\displaystyle
m_{1}\sum_{i=1}^{n}\Big{(}\max\left\\{\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{-}},\|z_{i}\|_{p_{i}(x),w_{i}}^{p_{i}^{+}}\right\\}+\mathcal{H}(k_{i}^{3})\max\left\\{\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{-}},\|z_{i}\|_{q_{i}(x),w_{i}}^{q_{i}^{+}}\right\\}\Big{)}}{\displaystyle\int_{\mathbb{R}^{N}}F(x,z_{1},\dots,z_{n})dx},\frac{r}{\displaystyle\int_{\mathbb{R}^{N}}\sup_{(\xi_{1},\dots,\xi_{n})\in
K(\frac{sr}{m_{0}^{\star}})}F(x,\xi_{1},\dots,\xi_{n})dx}\Bigg{[}$

system (1.1) admits at least three distinct weak solutions in $X$. ∎

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