# Positive Ground State Solutions to a Nonlocal Singular Elliptic Problem

Mustafa Avci  Mustafa Avci
Department of Finance and Management Science, Edwards School of Business,
University of Saskatchewan, Canada<EMAIL_ADDRESS>(Primary),
<EMAIL_ADDRESS>
###### Abstract.

In the present paper, we study the existence and uniqueness of solutions to
some nonlocal singular elliptic problem under Dirichlet boundary condition.
Problem is settled in Musielak-Sobolev spaces.

###### Key words and phrases:

Nonlocal elliptic problem, singular problem, Musielak-Orlicz-Sobolev spaces
2000 Mathematics Subject Classifications: 35J40; 35J60; 35J70

##  1\. Introduction

In this article, we are concerned with a nonlocal singular elliptic problem of
the form

$\begin{cases}\begin{array}[]{rlll}-\mathcal{A}\left(\int_{\Omega}\Phi(x,|\nabla
u|)dx\right)\mathrm{div}(a(x,|\nabla u|)\nabla u)&=&g(x)u^{-\gamma(x)}\text{
in }\Omega,\\\ u&>&0\text{ in }\Omega,\\\ u&=&0\text{ on
}\partial\Omega,\end{array}\end{cases}$ (1.1)

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ $(N\geq 3)$ with smooth
boundary $\partial\Omega$, $\gamma:\overline{\Omega}\rightarrow(0,1)$ is a
continuous function, $\mathcal{A}$ and $g$ are continuous functions.
The function $\varphi(x,t):=a(x,|t|)t$ is an increasing homeomorphism from
$\Omega\times\mathbb{R}$ onto $\mathbb{R}$. We want to remark that if we let
$a(x,t)=|t|^{p(x)-2}$, where $p(x)$ is a continuous function on
$\overline{\Omega}$ with $\inf_{x\in\overline{\Omega}}p(x)>1$, equation 1.1
turns into the well-known singular $p(x)$-Kirchhoff equation. If we
additionally consider the case $\mathcal{A}(t)=1$, equation 1.1 becomes
singular the $p(x)$-Laplace equation, a generalization of $p$-Laplace
equation, given by $\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u)$,
$1<p(x)<N$. Therefore, equation 1.1 particularly generalizes the problems
involving variable exponent. This kind of equations have been intensively
studied by many authors for the past two decades due to its significant role
in many fields of mathematics, such as in the study of calculus of variations,
partial differential equations [2, 17, 18], but also for their use in a
variety of physical and engineering contexts: the modeling of
electrorheological fluids [32], the analysis of Non-Newtonian fluids [36],
fluid flow in porous media [3], magnetostatics [14], image restoration [11],
and capillarity phenomena [8], see also, e.g., [4, 5, 6, 7, 9, 12, 13, 16, 23,
35] and references therein. Therefore, equation (1.1) may represent a variety
of mathematical models corresponding to certain phenomena:
For $\varphi(t):=p|t|^{p-2}t$;

  * •

Nonlinear elasticity: $\varphi(t)=\left(1+t^{2}\right)^{\alpha}-1,$
$\alpha>\frac{1}{2},$

  * •

Plasticity: $\varphi(t)=t^{\alpha}\left(\log\left(1+t\right)\right)^{\beta},$
$\alpha\geq 1,\beta>0,$

  * •

Generalized Newtonian fluids:
$\varphi(t)=\int_{0}^{t}s^{1-\alpha}\left(\sinh^{-1}s\right)^{\beta}ds,$
$0\leq\alpha\leq 1,\beta>0.$

For $\varphi(t)=\varphi(x,t):=p(x)|t|^{p(x)-2}t$;

  * •

There is a new model for image restoration given in [15]. In this model, main
aim is to recover an image, $u$, from an observed, noisy image, $u_{0}$, where
the two are related by $u_{0}=u+noise$. The proposed model incorporates the
strengths of the various types of diffusion arising from the minimization
problem

$E(u)=\int_{\Omega}\left[\left|\nabla
u\right|^{p\left(x\right)}+\lambda\left(u-u_{0}\right)^{2}\right]dx$

for $1\leq p\left(x\right)\leq 2$, where $\int_{\Omega}\left|\nabla
u\right|^{p\left(x\right)}dx$ is a regularizing term to remove the noise and
$\lambda\geq 0$.

Recently, there has been some paper dealing with nonlocal singular problems
see, e.g, [26, 27, 29, 33, 34] and references therein. However, to the best
knowledge of the author, problem (1.1) is not covered in the literature.

##  2\. Preliminaries

We use the theory of Orlicz spaces since problem 1.1 contains a nonhomogeneous
function $\varphi$ in the nonlinear differential operator
$\mathrm{div}(a(x,\cdot))$. Therefore, we start with some basic concepts of
Orlicz spaces. For more details, we refer the readers to the monographs [1,
24, 30, 31], and to the papers [19, 21, 22, 28].

The function $a(x,t):\Omega\times\mathbb{R}\rightarrow(0,\infty)$ is a
function such that the mapping
$\varphi(x,t):\Omega\times\mathbb{R}\to\mathbb{R}$, defined by

$\varphi(x,t)=\begin{cases}a(x,|t|)t&\text{ for }t\neq 0,\\\ 0,&\text{ for
}t=0,\end{cases}$ (2.1)

and for all $x\in\Omega$, $\varphi(x,\cdot):\mathbb{R}\to\mathbb{R}$ is an
odd, increasing homeomorphism. For the function $\varphi$ above, if we define

$\Phi(x,t)=\int_{0}^{t}\varphi(x,s)ds,\,\,\forall x\in\Omega,\,t\geq 0$ (2.2)

then the function $\Phi:\Omega\times[0,+\infty)\rightarrow[0,+\infty)$ is
called a generalized $N$-function if it satisfies the following conditions
(see e.g., [1, 30, 31]):

  * $(\Phi_{0})$

for almost all $x\in\Omega$, $\Phi(x,\cdot)$ is a $N$-function, i.e., convex,
nondecreasing and continuous function of $t$ such that, $\Phi(x,0)=0$,
$\Phi(x,t)>0$ for all $t>0$, and

$\lim_{t\rightarrow
0}\frac{\Phi(x,t)}{t}=0,\qquad\lim_{t\rightarrow\infty}\frac{\Phi(x,t)}{t}=+\infty$

  * $(\Phi_{1})$

$\Phi(\cdot,t)$ is a measurable function on $\Omega$ for all $t\geq 0$.

The set of all generalized $N$-functions is denoted by $N(\Omega)$. The
function $\bar{\Phi}$ defined by

$\bar{\Phi}(x,t)=\int_{0}^{t}\varphi^{-1}(x,s)ds,\,\,\forall x\in\Omega,t\geq
0$ (2.3)

is called the complementary (or conjugate) function to $\Phi$, where
$\bar{\Phi}$ satisfies the following

$\bar{\Phi}(x,t)=\sup_{s>0}\\{st-\Phi(x,s):~{}s\in\mathbb{R}\\},~{}\forall
x\in\Omega,t\geq 0$

It is well known that $\bar{\Phi}\in N(\Omega)$, and then the following Young
inequality holds

$st\leq\Phi(x,t)+\bar{\Phi}(x,s)~{}\text{for}~{}x\in\Omega~{}\text{and}~{}t,s\in\mathbb{R}.$
(2.4)

The function $\Phi$ allow us to define the Musielak-Sobolev spaces, also
called the generalized Orlicz spaces, by

$L^{\Phi}(\Omega)=\\{u:\Omega\rightarrow\mathbb{R}~{}\text{is
measurable};\exists\lambda>0~{}\text{such
that}~{}\int_{\Omega}\Phi(x,|u(x)|/\lambda)dx<+\infty\\}.$

Moreover, by $\Delta_{2}$-condition (see below), $L^{\bar{\Phi}}(\Omega)$ is
the dual space of $L^{\Phi}(\Omega)$, i.e.,
$(L^{\Phi}(\Omega))^{*}=L^{\bar{\Phi}}(\Omega)$.
In the sequel, we also use the following assumptions for $\Phi$:

$1<\varphi_{0}:=\inf_{t>0}\frac{t\varphi(x,t)}{\Phi(x,t)}\leq\frac{t\varphi(x,t)}{\Phi(x,t)}\leq\varphi^{0}:=\sup_{t>0}\frac{t\varphi(x,t)}{\Phi(x,t)}<\infty,~{}\forall
x\in\Omega,t\geq 0$ (2.5) $\operatorname*{ess\
inf}_{x\in\Omega}\Phi(x,t)>0,~{}\forall t>0$ (2.6) $\text{the
function}\,\,\,t\rightarrow\Phi(x,\sqrt{t})\,\,\,\text{is convex, }~{}\forall
x\in\Omega,t\geq 0$ (2.7)

By help of assumption (2.5), the Musielak-Sobolev spaces coincides the
equivalence classes of measurable functions $u:\Omega\rightarrow\mathbb{R}$
such that

$\int_{\Omega}\Phi(x,|u(x)|)dx<\infty$ (2.8)

and is equipped with the Luxembourg norm

$|u|_{\Phi}:=\inf\left\\{\mu>0:~{}\int_{\Omega}\Phi(x,|u(x)|/\lambda)dx\leq
1\right\\}$ (2.9)

For the Musielak-Orlicz spaces, Hölder inequality reads as follows (see
[1],[31])

$\int_{\Omega}uv\,dx\leq
2\|u\|_{L^{\Phi}(\Omega)}\|v\|_{L^{\bar{\Phi}}(\Omega)}\quad\text{ for all
}u\in L^{\Phi}(\Omega)\text{ and }v\in L^{\bar{\Phi}}(\Omega)$

The Musielak-Sobolev spaces $W^{1,\Phi}(\Omega)$ is the space defined by

$W^{1,\Phi}(\Omega):=\left\\{u\in L^{\Phi}(\Omega):~{}\frac{\partial
u}{\partial x_{i}}\in L^{\Phi}(\Omega),~{}i=1,2,...,N\right\\}$

under the norm

$\|u\|_{1,\Phi}:=|u|_{\Phi}+|\nabla u|_{\Phi}$ (2.10)

Now we introduce Musielak-Sobolev spaces with zero boundary traces
$W_{0}^{1,\Phi}(\Omega)$ as the closure of $C_{0}^{\infty}(\Omega)$ in
$W^{1,\Phi}(\Omega)$ under the norm $\|u\|_{1,\Phi}$. Moreover, by help of the
well-known Poincaré inequality, we can define an equivalent norm
$\|\cdot\|_{\Phi}$ on $W_{0}^{1,\Phi}(\Omega)$ by

$\|u\|_{\Phi}:=|\nabla u|_{\Phi}$ (2.11)

######  Remark 2.1.

  1. (1)

For the case $\Phi(x,t):=\Phi(t)$, we obtain $L^{\Phi}(\Omega)$ and
$W^{1,\Phi}(\Omega)$ called Orlicz spaces and Orlicz-Sobolev spaces,
respectively (see [24, 30, 31]).

  2. (2)

For the case $\Phi(x,t):=|t|^{p(x)}$, where $p(x)$ is a continuous function on
$\overline{\Omega}$ with $p(x)>1$, we replace $L^{\Phi}(\Omega)$ by
$L^{p(x)}(\Omega)$ and $W^{1,\Phi}(\Omega)$ by $W^{1,p(x)}(\Omega)$ and call
them variable exponent Lebesgue spaces and variable exponent Sobolev spaces,
respectively (see [1, 17, 18]).

######  Proposition 2.2 ([1]).

If (2.5)-(2.7) hold then the spaces $L^{\Phi}(\Omega)$ and
$W^{1,\Phi}(\Omega)$ are separable and reflexive Banach spaces.

######  Proposition 2.3 ([19, 28]).

Let define the modular $\rho(u):=\int_{\Omega}\Phi(x,|\nabla
u|)dx:W_{0}^{1,\Phi}(\Omega)\rightarrow\mathbb{R}$. Then for every $u_{n},u\in
W_{0}^{1,\varphi}(\Omega)$, we have

  * $(i)$

$\|u\|_{\Phi}^{\varphi^{0}}\leq\rho(u)\leq\|u\|_{\Phi}^{\varphi_{0}}$ if
$\|u\|_{\Phi}<1$

  * $(ii)$

$\|u\|_{\Phi}^{\varphi_{0}}\leq\rho(u)\leq\|u\|_{\Phi}^{\varphi^{0}}$ if
$\|u\|_{\Phi}>1$

  * $(iii)$

$\|u\|_{\Phi}\leq\rho(u)+1$

  * $(iv)$

$\|u_{n}-u\|_{\Phi}\rightarrow 0\Leftrightarrow\rho(u_{n}-u)\rightarrow 0$

  * $(v)$

$\|u_{n}-u\|_{\Phi}\rightarrow\infty\Leftrightarrow\rho(u_{n}-u)\rightarrow\infty$

We want to remark that Proposition 2.3 $(iv)-(v)$ mean that norm and modular
topology coincide on $L^{\Phi}(\Omega)$ provided $\Phi$ satisfies (2.5), which
enables that well-known $\Delta_{2}$-condition holds, i.e.,

$\Phi(x,2t)\leq M\Phi(x,t),\,\,\,\text{for all}\,\,\,x\in\Omega,t\geq 0$
(2.12)

where $M$ is a positive constant (see [28]).
Furthermore, if $\Psi,\Phi\in N(\Omega)$ and

$\Psi(x,t)\leq k_{1}\Phi(x,k_{2}t)+h(x),\,\,\,\text{for
all}\,\,\,x\in\Omega,t\geq 0$ (2.13)

holds, where $h\in L^{1}(\Omega)$ with $h(x)\geq 0$ a.e. $x\in\Omega$,
$k_{1},k_{2}$ are positive constants, then we have the following continuous
embeddings (see [30]):

  * $(i)$

$L^{\Phi}(\Omega)\hookrightarrow L^{\Psi}(\Omega)$

  * $(ii)$

$W^{1,\Phi}(\Omega)\hookrightarrow W^{1,\Psi}(\Omega)$

We also assume that the following condition hold for function $\Phi$.
For every $t>0$ there exists a constant $C_{t}>0$ such that

  * $(\Phi_{3})$

$C_{t}\leq\Phi(x,t)\leq C^{-1}_{t}$

for a.e. $x\in\Omega$.

######  Proposition 2.4 ([20]).

Assume that $\Omega$ is a bounded domain with smooth boundary
$\partial\Omega$. Then the embedding $W^{1,p(x)}(\Omega)\hookrightarrow
L^{r(x)}(\Omega)$ is compact provided $r,p\in C(\overline{\Omega})$ such that
$p^{-}>1$, $1\leq r(x)<p^{*}(x)$, where $p^{*}(x):=\frac{Np(x)}{N-p(x)}$ if
$p(x)<N$ and $p^{*}(x):=+\infty$ if $p(x)\geq N$.

######  Remark 2.5.

First, we note that for $t>1$ and $s>0$ it holds
$t^{\varphi_{0}}\Phi(x,s)\leq\Phi(x,ts)\leq t^{\varphi^{0}}\Phi(x,s)$. Indeed,
from the assumption (2.5), we have

$\varphi_{0}\leq\frac{z\varphi(x,z)}{\Phi(x,z)}\leq\varphi^{0},\,\,\,\forall
x\in\Omega,z\geq 0$

Considering that for almost all $x\in\Omega$, $\Phi(x,z)$ is a convex,
nondecreasing and continuous function of $z$, we can proceed as follows

$\int^{ts}_{s}\frac{\varphi_{0}}{z}dz\leq\int^{ts}_{s}\frac{\varphi(x,z)}{\Phi(x,z)}\leq\int^{ts}_{s}\frac{\varphi^{0}}{z}dz$

and hence

$t^{\varphi_{0}}\Phi(x,s)\leq\Phi(x,ts)\leq t^{\varphi^{0}}\Phi(x,s)$ (2.14)

Now, if we consider $(\Phi_{3})$ and the inequality (2.14) together, we can
obtain

$C_{t}t^{\varphi_{0}}\leq\Phi(x,st)+C$ (2.15)

Hence, if we consider (2.15) along with (2.13) where $\frac{1}{k_{1}}=C_{t}$,
$k_{2}=s$ and $h(x)=C=const\geq 0$, the Musielak-Sobolev space
$W^{1,\Phi}(\Omega)$ is continuously embedded in the variable Sobolev space
$W^{1,\varphi_{0}}(\Omega)$. On the other hand, $W^{1,\varphi_{0}}(\Omega)$ is
compactly embedded in the variable Lebesgue space $L^{r(x)}(\Omega)$ for all
$1\leq r(x)<\varphi_{0}^{*}:=\frac{N\varphi_{0}}{N-\varphi_{0}}$ with $r\in
C(\overline{\Omega})$. As a result, $W^{1,\Phi}(\Omega)$ is continuously and
compactly embedded in the variable Lebesgue space $L^{r(x)}(\Omega)$.

######  Remark 2.6.

The functional $\rho$ is from $C^{1}(W_{0}^{1,\Phi}(\Omega),\mathbb{R})$ with
the derivative

$\langle\rho^{\prime}(u),v\rangle=\int_{\Omega}a(x,|\nabla u|)\nabla
u\cdot\nabla vdx$

where $\langle\cdot,\cdot\rangle$ is the dual pairing between
$W_{0}^{1,\Phi}(\Omega)$ and its dual $(W_{0}^{1,\Phi}(\Omega))^{*}$(see
[28]).

The following Proposition generalizes the definition of convexity, and
therefore, we give a proof for the convenience.

######  Proposition 2.7.

Let $X$ be a vector space and let $I:X\rightarrow\mathbb{R}$. Then $I$ is
convex if and only if

$I((1-\lambda)u+\lambda v)<(1-\lambda)\theta+\lambda\beta,\,\,\,\,0<\lambda<1$
(2.16)

whenever $I(u)<\theta$ and $I(v)<\beta$, for all $u,v\in X$ and
$\theta,\beta\in\mathbb{R}$.

###### Proof.

Assume that functional $I:X\rightarrow\mathbb{R}$ is convex. Moreover, since
$I$ is a real-valued functional, there are real numbers
$\theta,\beta\in\mathbb{R}$ such that $I(u)<\theta$ and $I(v)<\beta$. Then

$I((1-\lambda)u+\lambda v)<(1-\lambda)I(u)+\lambda
I(v)<(1-\lambda)\theta+\lambda\beta,\,\,\,\,0<\lambda<1.$

On the other hand, assume that (2.16) holds. Since $I(u)<\theta$ and
$I(v)<\beta$, we can write, for all $\varepsilon>0$,

$I(u)<I(u)+\varepsilon:=\theta$ $I(v)<I(v)+\varepsilon:=\beta$

Therefore,

$I((1-\lambda)u+\lambda v)<(1-\lambda)I(u)+\lambda
I(v)+\varepsilon,\,\,\,\,0<\lambda<1$ (2.17)

If we consider that (2.17) holds for any $\varepsilon>0$, we conclude

$I((1-\lambda)u+\lambda v)\leq(1-\lambda)I(u)+\lambda I(v)$

∎

##  3\. The main results

######  Theorem 3.1.

Suppose that the following assumptions hold:

  * (G0)

$g(x)\in C^{1}(\overline{\Omega})$ is a nontrivial nonnegative function.

  * (A0)

$\mathcal{A}:(0,\infty)\rightarrow(0,\infty)$ is a continuous function and
satisfies the growth condition

$m_{1}t^{\alpha-1}\leq\mathcal{A}(t)\leq m_{2}t^{\alpha-1}$

where $m_{1},m_{2},\alpha$ are real numbers such that $m_{2}\geq m_{1}>1$ and
$\alpha>1$.

Then problem 1.1 has a positive ground state solution in
$W_{0}^{1,\Phi}(\Omega)$ with a negative energy level.

We define the functional $J:W_{0}^{1,\Phi}(\Omega)\rightarrow\mathbb{R}$
corresponding to problem 1.1 by

$J(u)=\widehat{\mathcal{A}}\left(\int_{\Omega}\Phi(x,|\nabla
u|)dx\right)-\int_{\Omega}\frac{g(x)|u|^{1-\gamma(x)}}{1-\gamma(x)}dx$

where $\widehat{\mathcal{A}}(t)=\int^{t}_{0}\mathcal{A}(s)ds$.

######  Definition 3.2.

A function $u$ is called a weak solution to problem 1.1 if $u\in
W_{0}^{1,\Phi}(\Omega)$ such that $u>0$ in $\Omega$ and

$\mathcal{A}\left(\int_{\Omega}\Phi(x,|\nabla
u|)dx\right)\int_{\Omega}a(x,|\nabla u|)\nabla u\cdot\nabla
vdx=\int_{\Omega}g(x)u^{-\gamma(x)}vdx$ (3.1)

for all $v\in W_{0}^{1,\Phi}(\Omega)$.

We would like to notice that due to singular term, the derivative operator
$J^{\prime}$ is not continuous on $W_{0}^{1,\Phi}(\Omega)$, that is, $J$ is
not Fréchet differentiable on $W_{0}^{1,\Phi}(\Omega)$. Therefore, we must
show that any global minimizer is in fact a solution to problem 1.1. To this
end, to obtain the main result given in Theorem 3.1, it is necessary to show
that Lemma 3.3 holds.

######  Lemma 3.3.

The functional $J$ attains the global minimizer in $W_{0}^{1,\Phi}(\Omega)$,
that is, there exists a function $u_{*}\in W_{0}^{1,\Phi}(\Omega)$ such that

$m=J(u_{*})=\inf_{u\in W_{0}^{1,\Phi}(\Omega)}J(u)<0$ (3.2)

###### Proof.

By (G0), (A0), Hölder inequality, Proposition 2.3 and the continuous
embeddings $W_{0}^{1,\Phi}(\Omega)\hookrightarrow L^{p(x)}(\Omega)$ and
$W_{0}^{1,\Phi}(\Omega)\hookrightarrow
L^{\frac{p(x)}{p(x)+\gamma(x)-1}}(\Omega)$, it follows

$\displaystyle|J(u)|$
$\displaystyle\leq\frac{m_{2}}{\alpha}\left(\int_{\Omega}\Phi(x,|\nabla
u|)dx\right)^{\alpha}+\frac{|g|_{\infty}}{1-\gamma^{+}}\int_{\Omega}|u|^{1-\gamma(x)}dx$
$\displaystyle\leq\frac{m_{2}}{\alpha}\|u\|_{\Phi}^{\alpha\varphi^{0}}+\frac{|g|_{\infty}}{1-\gamma^{+}}||u|^{1-\gamma(x)}|_{L^{\frac{p(x)}{1-\gamma(x)}}(\Omega)}|\mathbf{1}|_{L^{\frac{p(x)}{p(x)+\gamma(x)-1}}(\Omega)}$
$\displaystyle\leq\frac{m_{2}}{\alpha}\|u\|_{\Phi}^{\alpha\varphi^{0}}+\frac{c|g|_{\infty}}{1-\gamma^{+}}\|u\|^{1-\gamma^{-}}_{\Phi}<+\infty$

which shows that $J$ is well-defined on $W_{0}^{1,\Phi}(\Omega)$.
Denote $K:W_{0}^{1,\Phi}(\Omega)\rightarrow\mathbb{R}$ by
$K(u):=\widehat{\mathcal{A}}(\rho(u))$. Considering the fact that the
functional $\rho$ is of class $C^{1}(W^{1,\Phi}(\Omega),\mathbb{R})$ (see
Remark 2.6), and $\widehat{\mathcal{A}}$ is a continuous function, it is easy
to see that the composition functional $K$ is continuous on
$W_{0}^{1,\Phi}(\Omega)$. Further, by the well-known inequality

$|a^{p}-b^{p}|\leq|a-b|^{p},\,\,\,\text{for any real numbers}\,\,a,b\geq
0\,\,\text{and}\,\,0<p<1,$

we obtain

$\displaystyle|J(u)-J(v)|$
$\displaystyle\leq\left|\widehat{\mathcal{A}}(\rho(u))-\widehat{\mathcal{A}}(\rho(v))\right|+\frac{|g|_{\infty}}{1-\gamma^{+}}\int_{\Omega}||u|^{1-\gamma(x)}-|v|^{1-\gamma(x)}|dx$
$\displaystyle\leq|K(u)-K(v)|+\frac{|g|_{\infty}}{1-\gamma^{+}}\int_{\Omega}|u-v|^{1-\gamma(x)}dx$
$\displaystyle\leq|K(u)-K(v)|+\frac{|g|_{\infty}}{1-\gamma^{+}}||u-v|^{1-\gamma(x)}|_{L^{\frac{p(x)}{1-\gamma(x)}}(\Omega)}|\mathbf{1}|_{L^{\frac{p(x)}{p(x)+\gamma(x)-1}}(\Omega)}$
$\displaystyle\leq|K(u)-K(v)|+\frac{c|g|_{\infty}}{1-\gamma^{+}}\|u-v\|^{1-\gamma^{-}}_{\Phi}$

for any $u,v\in W_{0}^{1,\Phi}(\Omega)$. Therefore $J$ is continuous on
$W_{0}^{1,\Phi}(\Omega)$.
Let $u\in W_{0}^{1,\Phi}(\Omega)$. Then, applying the same steps as we did
above, it follows

$\displaystyle J(u)$
$\displaystyle\geq\frac{m_{1}}{\alpha}\left(\int_{\Omega}\Phi(x,|\nabla
u|)dx\right)^{\alpha}-\frac{|g|_{\infty}}{1-\gamma^{+}}||u|^{1-\gamma(x)}|_{L^{\frac{p(x)}{1-\gamma(x)}}(\Omega)}|\mathbf{1}|_{L^{\frac{p(x)}{p(x)+\gamma(x)-1}}(\Omega)}$
$\displaystyle\geq\frac{m_{1}}{\alpha}\|u\|_{\Phi}^{\alpha\varphi_{0}}-c\|u\|^{1-\gamma^{-}}_{\Phi}$
(3.3)

Since $\alpha\varphi_{0}>1-\gamma^{+}$, $J$ is coercive, namely,
$J(u)\rightarrow+\infty$ as $\|u\|_{\Phi}\rightarrow\infty$, and bounded
below.
Now, we shall show that $J$ is convex on $W_{0}^{1,\Phi}(\Omega)$. To this
end, using (A0) and considering the assumptions for $g$ and $\gamma$, we have

$\displaystyle J(u)$
$\displaystyle=\widehat{\mathcal{A}}\left(\int_{\Omega}\Phi(x,|\nabla
u|)dx\right)-\int_{\Omega}\frac{g(x)|u|^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\leq\frac{m_{2}}{\alpha}\max\\{\|u\|_{\Phi}^{\alpha\varphi_{0}},\|u\|_{\Phi}^{\alpha\varphi^{0}}\\}:=\theta$

and

$\displaystyle J(v)$
$\displaystyle=\widehat{\mathcal{A}}\left(\int_{\Omega}\Phi(x,|\nabla
v|)dx\right)-\int_{\Omega}\frac{g(x)|v|^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\leq\frac{m_{2}}{\alpha}\max\\{\|v\|_{\Phi}^{\alpha\varphi_{0}},\|v\|_{\Phi}^{\alpha\varphi^{0}}\\}:=\beta$

for all $u,v\in W_{0}^{1,\Phi}(\Omega)$. Since $\Phi$ is convex, so is
$\Theta(\cdot)=\int_{\Omega}\Phi(x,|\nabla\cdot|)dx$ (see [18]).
Since $\widehat{\mathcal{A}}$ satisfies (A0), it is a continuous and monotone
function on $(0,+\infty)$, and hence, its convexity follows. Therefore, for
$0<\lambda<1$, we have

$\widehat{\mathcal{A}}(\Theta((1-\lambda)u+\lambda
v))\leq(1-\lambda)\widehat{\mathcal{A}}(\Theta)+\lambda\widehat{\mathcal{A}}(\Theta)$

Therefore, considering the all pieces of information obtained above along with
(A0) and (F1), it follows

$\displaystyle J((1-\lambda)u+\lambda v))$
$\displaystyle=\widehat{\mathcal{A}}\left(\int_{\Omega}\Phi(x,|\nabla((1-\lambda)u+\lambda
v)|)dx\right)-\int_{\Omega}\frac{g(x)|(1-\lambda)u+\lambda
v|^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\leq(1-\lambda)\frac{m_{2}}{\alpha}\max\\{\|u\|_{\Phi}^{\alpha\varphi_{0}},\|u\|_{\Phi}^{\alpha\varphi^{0}}\\}+\lambda\frac{m_{2}}{\alpha}\max\\{\|v\|_{\Phi}^{\alpha\varphi_{0}},\|v\|_{\Phi}^{\alpha\varphi^{0}}\\}$
$\displaystyle\leq(1-\lambda)\theta+\lambda\beta$

Hence, by Proposition 2.7, $J$ is convex on $W_{0}^{1,\Phi}(\Omega)$.
As the functional $J$ is continuous, coercive and convex, it has a global
minimum belonging to $W_{0}^{1,\Phi}(\Omega)$, which in turn becomes a
solution to problem 1.1.
Let us denote

$m=\inf_{u\in W_{0}^{1,\Phi}(\Omega)}J(u)$

which is well-defined due to (3.3).
Now, applying the same arguments used in Remark 2.5, we can obtain that for
$t>0$ small enough and $s>0$, it holds $\Phi(x,ts)\leq
t^{\varphi_{0}}\Phi(x,s)$. Indeed, from assumption (2.5) and the properties of
$\Phi(x,z)$, we can proceed as follows

$\int^{s}_{ts}\frac{\varphi_{0}}{z}dz\leq\int^{s}_{ts}\frac{\varphi(x,z)}{\Phi(x,z)}dz$
$\Phi(x,ts)\leq t^{\varphi_{0}}\Phi(x,s)$

For $0\neq\phi\in W_{0}^{1,\Phi}(\Omega)$ and $0<t\in\mathbb{R}$ small enough,
it reads

$\displaystyle J(t\phi)$
$\displaystyle=\widehat{\mathcal{A}}\left(\int_{\Omega}\Phi(x,|\nabla
t\phi|)dx\right)-\int_{\Omega}\frac{g(x)|t\phi|^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\leq\frac{m_{2}}{\alpha}t^{\alpha\varphi_{0}}\left(\int_{\Omega}\Phi(x,|\nabla\phi|)dx\right)^{\alpha}-\frac{t^{1-\gamma^{+}}}{1-\gamma^{+}}\int_{\Omega}g(x)|\phi|^{1-\gamma(x)}dx$

Since $1-\gamma^{+}<\alpha\varphi_{0}$, we obtain that $J(t\phi)<0$. If we set
$t\phi=u$ with $\|u\|_{\Phi}<1$, we obtain that $m=\inf_{u\in
W_{0}^{1,\Phi}(\Omega)}J(u)<0$. On the other hand, if we take into account the
definition of $m$, there exists a minimizing sequence $(u_{n})$ of
$W_{0}^{1,\Phi}(\Omega)$ such that

$m=\lim_{n\rightarrow\infty}J(u_{n})<0$ (3.4)

Moreover, since $J(u_{n})=J(|u_{n}|)$ we may assume that $u_{n}\geq 0$. Due to
the coercivity of $J$, $(u_{n})$ must be bounded in $W_{0}^{1,\Phi}(\Omega)$
other wise we would have that $J(u_{n})\rightarrow+\infty$ as
$\|u_{n}\|_{\Phi}\rightarrow\infty$ which contradicts (3.4). Since
$W_{0}^{1,\Phi}(\Omega)$ is reflexive there exists a subsequence, not
relabelled, and $u_{*}\in W_{0}^{1,\Phi}(\Omega)$ such that

$u_{n}\rightharpoonup u_{*}$ in $W_{0}^{1,\Phi}(\Omega)$,

$u_{n}\rightarrow u_{*}$ in $L^{s(x)}(\Omega)$, $1\leq s(x)<p^{*}(x)$

$u_{n}(x)\rightarrow u_{*}(x)$ a.e. in $\Omega$.
Since $J$ is continuous and convex on $W_{0}^{1,\Phi}(\Omega)$, it is weakly
lower semi-continuous on $W_{0}^{1,\Phi}(\Omega)$. Therefore,

$\displaystyle m\leq J(u_{*})$
$\displaystyle=\widehat{\mathcal{A}}\left(\int_{\Omega}\Phi(x,|\nabla
u_{*}|)dx\right)-\int_{\Omega}\frac{g(x)|u_{*}|^{1-\gamma(x)}}{1-\gamma(x)}dx$
(3.5) $\displaystyle\leq\liminf_{n\rightarrow\infty}J(u_{n})=m$ (3.6)

which means

$m=J(u_{*})=\inf_{W_{0}^{1,\Phi}(\Omega)}J(u)<0$ (3.7)

∎

###### Proof.

(Proof of Theorem 3.1) Since $m=J(u_{*})<0=J(0)$, it must be $u_{*}\geq 0$,
$u_{*}\neq 0$. For $\phi\in W_{0}^{1,\Phi}(\Omega)$, $\phi\geq 0$ and $t>0$,
we have

$\displaystyle 0\leq$ $\displaystyle\liminf_{t\rightarrow
0}\frac{J(u_{*}+t\phi)-J(u_{*})}{t}$
$\displaystyle\leq\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla
u_{*}|)\nabla u_{*}\cdot\nabla\phi dx-\limsup_{t\rightarrow
0}\int_{\Omega}g(x)\frac{(u_{*}+t\phi)^{1-\gamma(x)}-u_{*}^{1-\gamma(x)}}{1-\gamma(x)}dx$

or

$\displaystyle\limsup_{t\rightarrow
0}\int_{\Omega}g(x)\frac{(u_{*}+t\phi)^{1-\gamma(x)}-u_{*}^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\leq\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla
u_{*}|)\nabla u_{*}\cdot\nabla\phi dx$ (3.8)

By the mean value theorem, there exists $\theta\in(0,1)$ such that

$\displaystyle\int_{\Omega}g(x)\frac{(u_{*}+t\phi)^{1-\gamma(x)}-u_{*}^{1-\gamma(x)}}{1-\gamma(x)}dx=$
$\displaystyle\int_{\Omega}g(x)(u_{*}+t\theta\phi)^{-\gamma(x)}\phi dx$ (3.9)

On the other hand, since we have

$(u_{*}+t\theta\phi)^{-\gamma(x)}\phi\geq 0,\,\,\,\forall x\in\Omega$

and

$(u_{*}+t\theta\phi)^{-\gamma(x)}\phi\rightarrow
u_{*}^{-\gamma(x)}\phi,\,\,as\,\,t\rightarrow 0,\,a.e.\,x\in\Omega$

we can apply Fatou’s lemma to (3.9), that is,

$\displaystyle\limsup_{t\rightarrow
0}\int_{\Omega}g(x)\frac{(u_{*}+t\phi)^{1-\gamma(x)}-u_{*}^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle\geq\liminf_{t\rightarrow
0}\int_{\Omega}g(x)\frac{(u_{*}+t\phi)^{1-\gamma(x)}-u_{*}^{1-\gamma(x)}}{1-\gamma(x)}dx$
$\displaystyle=\liminf_{t\rightarrow
0}\int_{\Omega}g(x)(u_{*}+t\theta\phi)^{-\gamma(x)}\phi dx$
$\displaystyle\geq\int_{\Omega}g(x)u_{*}^{-\gamma(x)}\phi dx\geq 0$ (3.10)

Thus, by (3.8) and (3.10) we can write

$\displaystyle\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla u_{*}|)\nabla
u_{*}\cdot\nabla\phi dx-\int_{\Omega}g(x)u_{*}^{-\gamma(x)}\phi dx$
$\displaystyle\geq 0,\,\,\,\forall\phi\in W_{0}^{1,\Phi}(\Omega),\phi\geq 0$
(3.11)

and hence, we obtain that function $u_{*}\in W_{0}^{1,\Phi}(\Omega)$ satisfies

$\displaystyle-\mathcal{A}(\rho(u_{*}))\mathrm{div}(a(x,|\nabla u_{*}|)\nabla
u_{*})$ $\displaystyle\geq 0\,\,\,\text{in}\,\,\Omega$ (3.12)

in the weak sense. Since $u_{*}\geq 0$ and $u_{*}\neq 0$, by the strong
maximum principle for weak solutions, we must have

$u_{*}(x)>0,\,\,\,\forall x\in\Omega$

Next, we show that $u_{*}\in W_{0}^{1,\Phi}(\Omega)$ satisfies (3.1). The
proof below has been adapted from one given in [25]. For given $\delta>0$,
define $\Lambda:[-\delta,\delta]\rightarrow(-\infty,\infty)$ by
$\Lambda(t)=J(u_{*}+tu_{*})$. Then $\Lambda$ achieves its minimum at $t=0$.
Thus,

$\displaystyle\frac{d}{dt}\Lambda(t)|_{t=0}=\frac{d}{dt}J(u_{*}+tu_{*})|_{t=0}=0$

or

$\displaystyle\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla u_{*}|)|\nabla
u_{*}|^{2}dx-\int_{\Omega}g(x)u_{*}^{1-\gamma(x)}dx=0$ (3.13)

Let us take $\phi\in W_{0}^{1,\Phi}(\Omega)$, and define $\Psi\in
W_{0}^{1,\Phi}(\Omega)$ such that
$\Psi:=(u_{*}+\varepsilon\phi)^{+}=\max\\{0,u_{*}+\varepsilon\phi\\}$,
$\varepsilon>0$. Clearly, $\Psi\geq 0$. If we replace $\Psi$ both in (3.11)
and (3.13), we have

$\displaystyle 0$
$\displaystyle\leq\mathcal{A}(\rho(u_{*}))\int_{\\{u_{*}+\varepsilon\phi\geq
0\\}}a(x,|\nabla u_{*}|)\nabla
u_{*}\cdot\nabla(u_{*}+\varepsilon\phi)dx-\int_{\\{u_{*}+\varepsilon\phi\geq
0\\}}g(x)u_{*}^{-\gamma(x)}(u_{*}+\varepsilon\phi)dx$
$\displaystyle=\mathcal{A}(\rho(u_{*}))\left(\int_{\Omega}-\int_{\\{u_{*}+\varepsilon\phi<0\\}}\right)a(x,|\nabla
u_{*}|)\nabla u_{*}\cdot\nabla(u_{*}+\varepsilon\phi)dx$
$\displaystyle-\left(\int_{\Omega}-\int_{\\{u_{*}+\varepsilon\phi<0\\}}\right)g(x)u_{*}^{-\gamma(x)}(u_{*}+\varepsilon\phi)dx$
$\displaystyle=\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla u_{*}|)|\nabla
u_{*}|^{2}dx-\int_{\Omega}g(x)u_{*}^{1-\gamma(x)}dx$
$\displaystyle+\varepsilon\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla
u_{*}|)\nabla u_{*}\cdot\nabla\phi
dx-\varepsilon\int_{\Omega}g(x)u_{*}^{-\gamma(x)}\phi dx$
$\displaystyle-\mathcal{A}(\rho(u_{*}))\int_{\\{u_{*}+\varepsilon\phi<0\\}}a(x,|\nabla
u_{*}|)\nabla
u_{*}\cdot\nabla(u_{*}+\varepsilon\phi)dx+\int_{\\{u_{*}+\varepsilon\phi<0\\}}g(x)u_{*}^{-\gamma(x)}(u_{*}+\varepsilon\phi)dx$
$\displaystyle=\varepsilon\left(\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla
u_{*}|)\nabla u_{*}\cdot\nabla\phi dx-\int_{\Omega}g(x)u_{*}^{-\gamma(x)}\phi
dx\right)$
$\displaystyle-\mathcal{A}(\rho(u_{*}))\int_{\\{u_{*}+\varepsilon\phi<0\\}}a(x,|\nabla
u_{*}|)\nabla
u_{*}\cdot\nabla(u_{*}+\varepsilon\phi)dx+\int_{\\{u_{*}+\varepsilon\phi<0\\}}g(x)u_{*}^{-\gamma(x)}(u_{*}+\varepsilon\phi)dx$
(3.14)
$\displaystyle\leq\varepsilon\left(\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla
u_{*}|)\nabla u_{*}\cdot\nabla\phi dx-\int_{\Omega}g(x)u_{*}^{-\gamma(x)}\phi
dx\right)$ (3.15)
$\displaystyle-\varepsilon\mathcal{A}(\rho(u_{*}))\int_{\\{u_{*}+\varepsilon\phi<0\\}}a(x,|\nabla
u_{*}|)\nabla u_{*}\cdot\nabla\phi dx$

Considering that $u_{*}>0$ and Lebesgue measure of the domain of integration
$\\{u_{*}+\varepsilon\phi<0\\}$ tends to zero as $\varepsilon\rightarrow 0$,
and (A0) it reads

$\mathcal{A}(\rho(u_{*}))\int_{\\{u_{*}+\varepsilon\phi<0\\}}a(x,|\nabla
u_{*}|)\nabla u_{*}\cdot\nabla\phi dx\rightarrow
0,\,\,\,\text{as}\,\varepsilon\rightarrow 0$

Moreover, considering that $a(x,\cdot)\in(0,\infty)$ and (A0), we can drop the
term

$-\mathcal{A}(\rho(u_{*}))\int_{\\{u_{*}+\varepsilon\phi<0\\}}a(x,|\nabla
u_{*}|)|\nabla u_{*}|^{2}dx$

in (3.14) since it is negative. Therefore, dividing (3.15) by $\varepsilon$
and letting $\varepsilon\rightarrow 0$, we obtain

$\displaystyle\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla u_{*}|)\nabla
u_{*}\cdot\nabla\phi dx-\int_{\Omega}g(x)u_{*}^{-\gamma(x)}\phi dx$
$\displaystyle\geq 0$ (3.16)

Considering that $\phi\in W_{0}^{1,\Phi}(\Omega)$ is arbitrary, (3.16) holds
for $-\phi$ as well. As a conclusion, we obtain

$\displaystyle\mathcal{A}(\rho(u_{*}))\int_{\Omega}a(x,|\nabla u_{*}|)\nabla
u_{*}\cdot\nabla\phi dx-\int_{\Omega}g(x)u_{*}^{-\gamma(x)}\phi dx$
$\displaystyle=0$ (3.17)

that is to say, $u_{*}\in W_{0}^{1,\Phi}(\Omega)$ is a weak solution to
problem 1.1. Additionally, since $J$ is coercive and bounded below on
$W_{0}^{1,\Phi}(\Omega)$, $u_{*}$ is a positive ground state solution to
problem 1.1, i.e., a solution with minimum action among all nontrivial
solutions. Additionally, since $J(u_{*})<0$ this solution has a negative
energy level. ∎

######  Theorem 3.4.

Suppose the conditions of Theorem 3.1 hold. Additionally, assume the following
conditions hold:

  * (A1)

$\mathcal{A}$ is bounded on $(0,\infty)$, i.e., for any $t\in(0,\infty)$,
there are real numbers $\underline{c},\overline{c}>0$ such that
$\underline{c}\leq\mathcal{A}(t)\leq\overline{c}$.

  * (a1)

There exists a real number $\underline{a}>0$ such that
$a(x,t)\geq\underline{a}>0$ holds for any $t\in\mathbb{R}$.

Then $u_{*}\in W_{0}^{1,\Phi}(\Omega)$ is the unique solution to problem 1.1.

###### Proof.

Let us assume $\upsilon_{*}$ is an another solution to problem 1.1. Then, from
(3.1), we have

$\displaystyle\mathcal{A}\left(\int_{\Omega}\Phi(x,|\nabla
u_{*}|)dx\right)\int_{\Omega}a(x,|\nabla u_{*}|)\nabla
u_{*}\cdot\nabla(u_{*}-\upsilon_{*})dx-\int_{\Omega}g(x)u_{*}^{-\gamma(x)}(u_{*}-\upsilon_{*})dx$
$\displaystyle-\mathcal{A}\left(\int_{\Omega}\Phi(x,|\nabla\upsilon_{*}|)dx\right)\int_{\Omega}a(x,|\nabla\upsilon_{*}|)\nabla\upsilon_{*}\cdot\nabla(u_{*}-\upsilon_{*})dx+\int_{\Omega}g(x)\upsilon_{*}^{-\gamma(x)}(u_{*}-\upsilon_{*})dx=0$

or

$\displaystyle\int_{\Omega}\left(\mathcal{A}(\rho(u_{*}))a(x,|\nabla
u_{*}|)\nabla
u_{*}-\mathcal{A}(\rho(\upsilon_{*}))a(x,|\nabla\upsilon_{*}|)\nabla\upsilon_{*}\right)\cdot\nabla(u_{*}-\upsilon_{*})dx$
(3.18)
$\displaystyle=\int_{\Omega}g(x)(u_{*}^{-\gamma(x)}-\upsilon_{*}^{-\gamma(x)})(u_{*}-\upsilon_{*})dx$
(3.19)

For $\alpha\in(0,1)$ and $x,y\geq 0$, we have the elementary inequality

$\displaystyle(x^{-\alpha}-y^{-\alpha})(x-y)\leq 0$ (3.20)

Moreover, by Lemma 2.4 given in [10], we have the following inequality: for
any $k,l>0$, there exists a positive constant $C(\delta)$,
$\delta=\min\\{1,a_{0},k,l\\}$, such that

$(ka(|\xi|)\xi-la(|\eta|)\eta)\cdot(\xi-\eta)\geq
C(\delta)\Phi(|\xi-\eta|)\quad\forall\xi,\eta\in\mathbb{R}^{N}$ (3.21)

holds, provided that (A1) and (a1) hold. Thus, if we apply (3.21) and (3.20)
to the lines (3.18) and (3.19) respectively, we obtain

$\displaystyle 0\leq$ $\displaystyle C(\delta)\int_{\Omega}\Phi(x,|\nabla
u_{*}-\nabla\upsilon_{*}|)dx\leq 0$

or

$\displaystyle\rho(u_{*}-\upsilon_{*})=0$

Therefore, by Proposition 2.3, we have

$\displaystyle
0\leq\min\\{\|u_{*}-\upsilon_{*}\|^{\varphi_{0}}_{\Phi},\|u_{*}-\upsilon_{*}\|^{\varphi^{0}}_{\Phi}\\}\leq\rho(u_{*}-\upsilon_{*})=0$

which means that

$\displaystyle\|u_{*}-\upsilon_{*}\|_{\Phi}=0$

Thus, we have $u_{*}=\upsilon_{*}$ in $W_{0}^{1,\Phi}(\Omega)$, that is,
$u_{*}$ is the unique solution to problem 1.1. ∎

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