Existence results for double phase obstacle problems with variable exponents

Abstract

In this paper, we introduce a new class of the approximating problems corresponding to a quasilinear obstacle equations, which involves a general variable exponents elliptic operator in divergence form, called double phase obstacle operator with variable exponents. Based on the mountain pass theorem, tools from non-smooth analysis, and some suitable assumptions, we prove the existence of weak solutions.

1 Introduction

Let $\varOmega \subset {\mathbb {R}}^{N}$ be a bounded domain with Lipschitz boundary $\partial \varOmega$. In this paper, we are addressed in proving the existence of weak solutions to the following variable exponents quasilinear equation:

$$\begin{aligned} ( {\mathcal {P}} ) \, {\left\{ \begin{array}{ll} - \, {\text {div}} A( x, \nabla u ) + | u |^{\alpha ( x ) - 2} u = \lambda \, b( x ) | u |^{\beta ( x ) - 2} u &{} {\text {in}} \quad \varOmega , \\ \, u( x ) \, \le \varPhi ( x ) \, &{} {\text {in}} \quad \varOmega ,\\ \, u( x ) \, = \, 0 &{} {\text {on}} \quad \partial \varOmega . \end{array}\right. } \end{aligned}$$

Where $\lambda > 0$ is a parameter, $A: {\mathbb {R}} \times {\mathbb {R}}^{N} \longrightarrow {\mathbb {R}}^{N}$ admits a potential ${\mathcal {A}},$ with respect to its second variable $\zeta$, satisfying the following assumptions:

$( {\mathcal {A}}_{1} )$ The potential ${\mathcal {A}} = {\mathcal {A}} ( x, \epsilon )$ is a continuous function in $\varOmega \times \varOmega ,$ with continuous derivative with respect to $\zeta , \, A = \partial _{\epsilon } {\mathcal {A}} ( x, \epsilon ),$ and verifies:

• (i) ${\mathcal {A}} (x, \cdot )$ is strictly convex in $\varOmega$ for all $x \in \varOmega$;

• (ii) ${\mathcal {A}} ( x, 0 ) = 0$ and ${\mathcal {A}} ( x, \zeta ) = {\mathcal {A}} ( x, - \zeta )$, for all $( x, \zeta ) \in \varOmega \times \varOmega$

• (iii) There exist positive constants $c_{1}, \, c_{2}$ and variable exponents $p( \cdot ), \,q( \cdot ) : \varOmega \rightarrow {\mathbb {R}}$ such that for all $( x, \zeta ) \in \varOmega \times \varOmega$

  $$\begin{aligned} A( x, \zeta ) \,.\, \zeta \ge \, {\left\{ \begin{array}{ll} c_{1} \, | \zeta |^{p( x )} ;&{} {\text {if}} \quad | \zeta | \gg 1 , \\ c_{1} \, | \zeta |^{q( x )} ;&{} {\text {if}} \quad | \zeta | \ll 1 , \end{array}\right. } \end{aligned}$$

  and

  $$\begin{aligned} | A( x, \zeta ) | \le \, {\left\{ \begin{array}{ll} c_{2} \, | \zeta |^{p( x ) - 1} ; &{} {\text {if}} \quad | \zeta | \gg 1 , \\ c_{2} \, | \zeta |^{q( x ) - 1} ; &{} {\text {if}} \quad | \zeta | \ll 1 , \end{array}\right. } \end{aligned}$$

• (iv) $1 \ll p( \cdot ) \ll q( \cdot ) \ll {\text {min}} \{ N, \, p^{*} ( \cdot ) \},$ and $p( \cdot ), \, q( \cdot )$ are Lipschitz continuous in ${\mathbb {R}}^{N},$ satisfy

  $$\begin{aligned} \frac{q( \cdot )}{p( \cdot )} < 1 + \frac{1}{N}, \end{aligned}$$

  where

  $$\begin{aligned} p^{*} ( x ) = \, {\left\{ \begin{array}{ll} \frac{N p( x )}{N - p( x )} ;&{} {\text {if}} \quad p( x ) < N , \\ \infty ;&{} {\text {if}} \quad p( x ) \ge N , \end{array}\right. } \end{aligned}$$

• (v) $A( x, \zeta ) \,.\, \zeta \le s( \cdot ) \, {\mathcal {A}} ( x, \zeta )$ for any $( x, \zeta ) \in \varOmega \times \varOmega ,$ where $s( \cdot ): \varOmega \rightarrow {\mathbb {R}}$ is Lipschitz continuous and satisfies $q( \cdot ) \le s( \cdot ) \ll p^{*} ( \cdot ).$ $( {\mathcal {A}}_{2} )$ ${\mathcal {A}}$ is uniformly convex, that is, for any $0< \epsilon < 1,$ there exists $\delta ( \epsilon ) \in ( 0, \,1 )$ such that $| u - v | \le \epsilon \, {\text {max}} \{ | u |, \, | v | \}$ or

  $$\begin{aligned} {\mathcal {A}} \bigg ( x, \frac{u + v }{2} \bigg ) \le \frac{1}{2} \, ( 1 - \delta ( \epsilon ) ) \, ( {\mathcal {A}} ( x, u ) + {\mathcal {A}} ( x, v ) ) \end{aligned}$$

  for any $x, \, u, \, v \in \varOmega$

• (vi) $\varPhi : \varOmega \longrightarrow {\mathbb {R}}^{+}$ is a given function satisfy: $\varPhi \in L^{q( x )} ( \varOmega )$

• (viii) $w \in L^{r( \cdot )} ( \varOmega ), \, w > 0$ a.e in $\varOmega , \, 1 \ll r( x ) \ll \infty ,$ and

  $$\begin{aligned} r'( x ) \le \frac{p^{*} ( x )}{\gamma ( x )}, \,\,\, \forall x \in \varOmega , \end{aligned}$$

  where $r'( x )$ is the conjugate function of r(x),  namely $\frac{1}{r( x )} + \frac{1}{r'( x )} = 1$ and $\gamma ( \cdot )$ is Lipschitz continuous satisfy $\alpha \le \gamma ( \cdot ) \ll p^{*} ( \cdot ).$

• (ix) $1 \ll \beta ( x ) \le \beta ^{+} < \alpha ^{-}, \,\, b \in L^{r^{*} ( \cdot )} ( \varOmega ), \,\, b > 0$ a.e in $\varOmega , \,\, 1 \ll r^{*} ( x ) \ll \infty ,$ and

  $$\begin{aligned} \alpha ( x ) \le \frac{r^{*} ( x )}{r^{*} ( x ) - 1} \, \beta ( x ) \le p^{*} ( x ) \,\,\, \forall x \in \varOmega . \end{aligned}$$

Recently, the existence of solutions to nonlinear problems with non-homogeneous structures have received much attention, particularly the existence of solutions to double phase problems with variable exponents. These operators are the natural extension of the classical double phase problems when p and q are constants. For example, in the special case $p = q =$ constant and using the surjectivity theorem, multivalued mapping Kluge’s fixed point principle and tools from nonsmooth analysis, Zeng et al. in [26] showed the existence of at least one solution of the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} - \, {\text {div}} ( | \nabla u |^{p - 2} \nabla u + \mu ( x ) | \nabla u |^{q - 2} \nabla u ) + \partial j ( x, u ) \ni f( x ) &{} {\text {in}} \quad \varOmega , \\ \, u( x ) \, = \, 0 &{} {\text {on}} \quad \partial \varOmega , \\ {\mathcal {T}} ( u ) \le U( x ) \end{array}\right. } \end{aligned}$$

which is described by Clarke’s generalized gradient, where $1< p< q < N, \, \mu : {\overline{\varOmega }} \rightarrow {\mathbb {R}}^{+}$, ${\mathcal {T}}, \, U: W_{0}^{1, {\mathcal {H}}} \rightarrow {\mathbb {R}}$ are functions satisfying same appropriate condition and $W_{0}^{1, {\mathcal {H}}}$ is a subspace of the Sobolev–Musielak–Orlicz space $W^{1, {\mathcal {H}}}$ generated by the $\varphi$-function ${\mathcal {H}} ( x, t ) = | t |^{p} + \mu ( x ) \, | t |^{q}.$ Next, the same authors in [27] introduced a family of the approximating problems corresponding to an elliptic obstacle problem with a double phase phenomena and a multivalued reaction convection term, namely:

$$\begin{aligned} ( {\mathcal {P}}_{1} ) {\left\{ \begin{array}{ll} - \, \text{ div } ( | \nabla u |^{p - 2} \nabla u + \mu ( x ) | \nabla u |^{q - 2} \nabla u ) \in f( x, u, \nabla u ) &{} {\text {in}} \quad \varOmega , \\ \, u( x ) \, \le \varPhi ( x ) &{} {\text {on}} \quad \varOmega , \\ \, u( x ) \, = \, 0 &{} {\text {on}} \quad \partial \varOmega , \end{array}\right. } \end{aligned}$$

and they established the following convergence relation

$$\begin{aligned} \emptyset \ne w - \lim _{n \rightarrow \infty } \sup S_{n} = s - \lim _{n \rightarrow \infty } \sup S_{n} \subset S, \end{aligned}$$

where, $w - \lim _{n \rightarrow \infty } \sup S_{n}$ and $s- \lim _{n \rightarrow \infty } \sup S_{n}$ denote the weak and the strong Kuratowski upper of $S_{n}$ respectively, with S the solution set of the obstacle problem and $S_{n}$ the solution sets of approximating problems. For more results, we refer the reader to [1, 2, 5,6,7, 28] and the references given there.

When $\varPhi \equiv + \infty ,$ the problem $( {\mathcal {P}}_{1} )$ becomes the following double phase problem with multivalued convection term.

$$\begin{aligned} {\left\{ \begin{array}{ll} - \, \text{ div } ( | \nabla u |^{p - 2} \nabla u + \mu ( x ) | \nabla u |^{q - 2} \nabla u ) \in f( x, u, \nabla u ) &{} {\text {in}} \quad \varOmega , \\ \, u( x ) \, = \, 0 &{} {\text {on}} \quad \partial \varOmega . \end{array}\right. } \end{aligned}$$

Moreover, when f is a single-valued function, the foregoing problem becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} - \, \text{ div } ( | \nabla u |^{p - 2} \nabla u + \mu ( x ) | \nabla u |^{q - 2} \nabla u ) = f( x, u, \nabla u ) &{} {\text {in}} \quad \varOmega , \\ \, u( x ) \, = \, 0 &{} {\text {on}} \quad \partial \varOmega . \end{array}\right. } \end{aligned}$$

which was recently treated by Gasiński, and Winkert in [12]. Let us mention some relevant papers in this direction. Zhikov in [30] describe models of strongly anisotropic materials by treating the functional

$$\begin{aligned} u \longmapsto \int _{\varOmega } ( | \nabla u |^{p} + \mu ( x ) \, | \nabla u |^{q} ) \,\, dx. \end{aligned}$$

(1)

Li et al. in [14] proved the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to the general nonlinear elliptic equations in Musielak–Orlicz spaces. Moreover, they also obtain the equivalence of entropy solutions and renormalized solutions. For a deeper comprehension, we refer the reader to [4, 9, 15,16,17,18] and the references therein for more background.

Moving on to another novel aspect; the double phase problem with variable exponents that few author consider. Ragusa and Tachikawa in [19,20,21,22] and reference therein, are the first ones who have achieved the regularity theory for minimizers of (1) with variable exponents. Moreover, in [25] Tachikawa, provides the Hölder continuity up to the boundary of minimizers of so-called double phase functional with variable exponents, under suitable Dirichlet boundary conditions. Recently, Zhang and Rădulescu in [29] proved the existence of multiple solution for the quasilinear equation

$$\begin{aligned} \text{ div } A( x, \nabla u ) + V( x ) | u |^{\alpha ( x ) - 2} u = f( x, u ) \,\,\, \text{ in } \,\,\, {\mathbb {R}}^{N}, \end{aligned}$$

which involves a general variable exponent elliptic operator in divergence form. This type of problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like $| \zeta |^{q( x ) - 2} \zeta$ for small $| \zeta |$ and like $| \zeta |^{p( x ) - 2} \zeta$ for large $| \zeta |$ where $1< \alpha ( \cdot ) \le p( \cdot )< q( \cdot ) < N.$ We refer to other methods to solve this type of problems which can be found in the work of Shi, Rădulescu, Repovš, and Zhang [24] and the references therein. Many results have been obtained concerning the application of this type of problem in different sectors, such as the study of fluid filtration in porous media, constrained heating, elastoplasticity, optimal control, financial mathematics, etc. Readers may refer to [8, 13, 23, 31, 32] and the references therein for more background of applications.

The novelty of our paper is the fact that we combine several different phenomena in one problem. More precisely, our problem $( {\mathcal {P}} )$ contains: Quasilinear equation; which involves a general variable exponents elliptic operator in divergence form, an obstacle restriction, and double phase operators; the reason why it is called double phase, is that (1) is defined by the fact that the energy density changes its ellipticity and growth properties depending on the point in the domain. To be specific, its behavior depends on the values of the weight function $\mu ( \cdot ).$ Actually, on the set $\{ x \in \varOmega / \mu ( x ) = 0 \}$ it will be composed by the gradient of order $p( \cdot )$ and on the set $\{ x \in \varOmega / \mu ( x ) \ne 0 \}$ it is the gradient of order $q( \cdot ).$

To the best of our knowledge, no previous research has investigated the double phase obstacle operator with variable exponents given in the general form $({\mathcal {P}})$. Besides, we address the challenges that come about due to the non-homogeneities of the growths, and the presence of several non-linear terms.

The rest of our paper is organized as follows. In Sect. 2, we briefly review some properties of the Lebesgue spaces with variable exponents and the Theory of Sobolev–Orlicz spaces with variable exponents. In Sect. 3, we establish the existence results of weak solutions to the problem $( {\mathcal {P}} )$ using the mountain pass Theorem, tools from non-smooth analysis and some suitable assumptions.

2 Notations and basic properties

In order to discuss the problem $( {\mathcal {P}} )$, we need some facts on spaces $L^{q( x )} ( \varOmega )$ and $W_{0}^{1, q( x )} ( \varOmega )$ where $\varOmega$ is an open subset of ${\mathbb {R}}^{N}$ which are called the Lebesgue spaces with variable exponents and the Sobolev spaces with variable exponents setting. For this reason, we will recall some properties involving the above spaces, which can be found in [3, 4, 10, 11] and references therein.

Let $\varOmega$ be a bounded open subset of ${\mathbb {R}}^{N} \, ( N \ge 2 )$, we define the Lebesgue space with variable exponent $L^{q(\cdot )} ( \varOmega )$ as the set of all measurable function $u : \varOmega \longmapsto {\mathbb {R}}$ for which the convex modular

$$\begin{aligned} \rho _{q(\cdot )} ( u ) = \int _{\varOmega } |\, u( x )\,|^{q( x )} \,\, dx, \end{aligned}$$

is finite. If the exponent is bounded, i.e if $q^{+} = ess \,sup \{ \, q( x ) / x \in \varOmega \, \} < + \infty ,$ then the expression

$$\begin{aligned} ||\, u \,||_{q(\cdot )} = \inf \{ \, \lambda > 0: \, \rho _{q(\cdot )} \bigg ( \frac{u}{\lambda } \bigg ) \le 1 \, \}, \end{aligned}$$

defines a norm in $L^{q(\cdot )} ( \varOmega )$, called the Luxemburg norm.

The space $( L^{q(\cdot )} ( \varOmega ), \, ||\,.\,||_{q(\cdot )} )$ is a separable Banach space. Moreover, if $1< q^{-} \le q^{+} < +\infty ,$ then $L^{q(\cdot )} ( \varOmega )$ is uniformly convex, where $q^{-} = ess\, inf \{ \, q( x ) / x \in \varOmega \, \},$ hence reflexive, and its dual space is isomorphic to $L^{q'(\cdot )} ( \varOmega )$ where $\frac{1}{q( x )} + \frac{1}{q'( x )} = 1.$

Finally, we have the Hölder type inequality:

$$\begin{aligned} \bigg |\, \int _{\varOmega } u\, v \,\, dx \, \bigg | \le \bigg ( \, \frac{1}{q^{-}} + \frac{1}{( q^{'} )^{-}} \bigg ) \, ||\, u\,||_{q(\cdot )} ||\, v\,||_{q^{'} (\cdot )}, \end{aligned}$$

(2)

for all $u \in L^{q(\cdot )} ( \varOmega )$ and $v \in L^{q' (\cdot )} ( \varOmega ).$

We define the variable exponents Sobolev space by

$$\begin{aligned} W^{1, q( x )} ( \varOmega ) = \{ \, u \in L^{q( x )} ( \varOmega ) \,\, \text{ and } \,\, |\, \nabla u\,| \in L^{q( x )} ( \varOmega ) \,\}, \end{aligned}$$

with the norm

$$\begin{aligned} ||\, u\,||_{W^{1, q( x )} ( \varOmega )} = ||\, u\,||_{L^{q( x )} ( \varOmega )} + ||\, \nabla u \Vert |_{L^{q( x )} ( \varOmega )} \,\, \forall u \in W^{1, q( x )} ( \varOmega ). \end{aligned}$$

We denote by $W^{1, q( x )}_{0} ( \varOmega )$ the closure of $C_{0}^{\infty } ( \varOmega )$ in $W^{1, q( x )} ( \varOmega ),$ and we define the Sobolev exponent by $q^{*} ( x ) = \frac{N \, q( x )}{N - q( x )}$ for $q( x ) < N.$

2.1 Theory of Sobolev–Orlicz spaces with variable exponents

In this part, we recall some relevant definitions and properties that will help us in our analysis. For the convenience of the readers and for the sake of completeness, we recall the proofs of some results.

Definition 1

We define the following real valued linear space

$$\begin{aligned} L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega ) = \{ u / u = v + w, \, v \in L^{p( \cdot )} ( \varOmega ), \, w \in L^{q( \cdot )} ( \varOmega ) \, \}, \end{aligned}$$

which is endowed with the norm

$$\begin{aligned} | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega )}= & {} \inf \,\{ | v |_{L^{p( \cdot )} ( \varOmega )} + | w |_{L^{q( \cdot )} ( \varOmega )} \, / \, v \in L^{p( \cdot )} ( \varOmega ), \, w \in L^{q(\cdot )} ( \varOmega ),\nonumber \\ u= & {} v +w \,\}. \end{aligned}$$

(3)

We also defined the linear space

$$\begin{aligned} L^{p( \cdot )} ( \varOmega ) \cap L^{q( \cdot )} ( \varOmega ) = \{ \, u \,/\, u \in L^{p( \cdot )} ( \varOmega ) \,\, \text{ and } \,\, u \in L^{q( \cdot )} (\varOmega ) \,\}, \end{aligned}$$

which is endowed with the norm

$$\begin{aligned} | u |_{L^{p( \cdot )} ( \varOmega ) \cap L^{q( \cdot )} (\varOmega )} = \max \, \{ \, | u |_{L^{p( \cdot )} ( \varOmega )}, \, | u |_{L^{q( \cdot )} ( \varOmega )} \, \}. \end{aligned}$$

We denote

$$\begin{aligned} {\mathcal {D}}_{u} = \{ x \in \varOmega \, /\, | u( x ) | > 1 \} \,\,\, \text{ and } \,\,\, {\mathcal {D}}^{c}_{u} = \{ x \in \varOmega \, / \, | u( x ) | \le 1 \}. \end{aligned}$$

Proposition 1

Assume $( {\mathcal {A}}_{1} )$—(iv). Let $\varOmega \subset {\mathbb {R}}^{N}$ and $u \in L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega ).$ Then, the following properties hold:

• (a) If $\varOmega ' \subset \varOmega$ is such that $| \varOmega ' | < + \infty ,$ then $u \in L^{p( \cdot )} ( \varOmega ' ).$

• (b) If $\varOmega ' \subset \varOmega$ is such that $u \in L^{\infty } ( \varOmega ' ),$ then $u \in L^{q( \cdot )} ( \varOmega ' ).$

• (c) $| {\mathcal {D}}_{u} | < + \infty .$

• (d) $u \in L^{p( \cdot )} ( {\mathcal {D}}_{u}) \cap L^{q( \cdot )} ( {\mathcal {D}}^{c}_{u}).$

• (e) The infimum in (3) is attained.

• (f) If $B \subset \varOmega ,$ then, $| u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega )} \le | u |_{L^{p( \cdot )} ( B ) + L^{q( \cdot )} ( B )} + | u |_{L^{p( \cdot )} ( \varOmega \backslash B ) + L^{q( \cdot )} ( \varOmega \backslash B )}.$

• (g) We have

  $$\begin{aligned}&\max \left\{ \frac{1}{1 + 2 \, | {\mathcal {D}}_{u}|^{\frac{1}{p( \zeta )} - \frac{1}{q( \zeta )}}} \, | u |_{L^{p( \cdot )} ( {\mathcal {D}}_{u})}, \, c\, \min \left\{ | u |_{L^{q( \cdot )} ( {\mathcal {D}}^{c}_{u})}, \, | u |_{L^{q( \cdot )} ( {\mathcal {D}}^{c}_{u} )}^{\frac{q( \zeta )}{p( \zeta )}} \, \right\} \, \right\} \\&\quad \le | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega )} \le | u |_{L^{p( \cdot )} ( {\mathcal {D}}_{u} )} + | u |_{L^{q( \cdot )} ( {\mathcal {D}}^{c}_{u})}, \end{aligned}$$

  where, $\zeta \in \varOmega$ and c is a small positive constant.

Proof

Let $v \in L^{p( \cdot )} ( \varOmega )$ and $w \in L^{q( \cdot )} ( \varOmega )$ be such that $u = v + w,$ then $v \in L^{p^{'}( \cdot )} ( \varOmega ' )$ and $w \in L^{q^{'} ( \cdot )} ( \varOmega ' ).$

• (a) To show that $u \in L^{p( \cdot )} ( \varOmega ' ),$ it is enough to show that $w \in L^{p( \cdot )} ( \varOmega ' )$. And by Young’s inequality, we get the results.

• (b) To prove $u \in L^{q( \cdot )} ( \varOmega ' ),$ it is sufficient to prove that $v \in L^{q( \cdot )} ( \varOmega ' )$. For that, we have

  $$\begin{aligned} \int _{\varOmega '} | v( x ) |^{q( x )} \,\, dx&= \int _{\varOmega '} | v( x ) |^{q( x ) - p( x )} | v( x ) |^{p( x )} \,\, dx\\ {}&\le ( 1 + | \sup v | )^{q^{+} - p^{-}} \int _{\varOmega '} | v( x ) |^{p( x )} \,\, dx < + \infty . \end{aligned}$$

  Thus, $v \in L^{q( \cdot )} ( \varOmega ' ).$ Hence, $u \in L^{q( \cdot )} ( \varOmega ' ).$

• (c) We use the fact that $1 < | u | \le | v | + | w |$ implies that $| v | \ge \frac{1}{2}$ or $| w | \ge \frac{1}{2}$ and for all $x \in \varOmega ,$ we get

  $$\begin{aligned} + \infty > \int _{\varOmega } | u |^{p( x )} + | w |^{q( x )} \,\, dx \ge \bigg | \frac{1}{2} \bigg |^{p^{+} + q^{+}} \, | {\mathcal {D}}_{u} |. \end{aligned}$$

  Hence, the result.

• (d) From the assumptions $(a)-(c)$ we get the result.

• (e) Let $u \in L^{p(\cdot )} ( \varOmega ) + L^{q(\cdot )} ( \varOmega ),$ we consider a minimizing sequence for u,  namely $v_{n} \in L^{p( \cdot )} ( \varOmega )$ and $w_{n} \in L^{q( \cdot )} ( \varOmega )$ such that $u = v_{n} + w_{n}$ and

  $$\begin{aligned} \lim _{n \rightarrow + \infty } \big ( | v_{n} |_{L^{p( \cdot )} ( \varOmega )} + | w_{n} |_{L^{q( \cdot )} ( \varOmega )} \big ) = | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega )}. \end{aligned}$$

  Then, by the reflexibility of $L^{p( \cdot )} ( \varOmega )$ and $L^{q( \cdot )} (\varOmega ),$ there exist $v_{0} \in L^{p( \cdot )} ( \varOmega )$ and $w_{0} \in L^{q( \cdot )} ( \varOmega )$ such that $v_{n} \rightharpoonup v_{0}$ in $L^{p( \cdot )} ( \varOmega )$ and $w_{n} \rightharpoonup w_{0}$ in $L^{q( \cdot )} ( \varOmega ).$ By lower continuity, we have

  $$\begin{aligned} | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} (\varOmega )}&= \lim _{n \rightarrow + \infty } \big ( | v_{n} |_{L^{p( \cdot )} ( \varOmega )} + | w_{n} |_{L^{q( \cdot )} ( \varOmega )} \big ) \\ {}&\ge \lim _{n \rightarrow + \infty } \inf | v_{n} |_{L^{p( \cdot )} ( \varOmega )} + \lim _{n \rightarrow + \infty } \inf | w_{n} |_{L^{q( \cdot )} ( \varOmega )} \\&= | v_{0} |_{L^{p( \cdot )} ( \varOmega )} + | w_{0} |_{L^{q( \cdot )} ( \varOmega )}. \end{aligned}$$

  According to the Definition 1, we get

  $$\begin{aligned} | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} (\varOmega )} = | v_{0} |_{L^{p( \cdot )} ( \varOmega )} + | w_{0} |_{L^{q( \cdot )} ( \varOmega )}. \end{aligned}$$

• (f) See Proposition 2.2 in [4].

• (g) From (d) and the Definition 1, we notice that

  $$\begin{aligned} | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q(\cdot )} ( \varOmega )} \le | u |_{L^{p(\cdot )} ( {\mathcal {D}}_{u}^{c} )} + | u |_{L^{q( \cdot )} ( {\mathcal {D}}_{u}^{c} )}. \end{aligned}$$

  By (e), we obtain that

  $$\begin{aligned} | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q(\cdot )} ( \varOmega )} \ge | u |_{L^{p( \cdot )} ( {\mathcal {D}}_{u} ) + L^{q(\cdot )} ( {\mathcal {D}}_{u} )}, \end{aligned}$$

  and

  $$\begin{aligned} | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q(\cdot )} ( \varOmega )} \ge | u |_{L^{p( \cdot )} ( {\mathcal {D}}^{c}_{u} ) + L^{q(\cdot )} ( {\mathcal {D}}^{c}_{u} )}. \end{aligned}$$

  By (2), we deduce that

  $$\begin{aligned} | u |_{L^{p( \cdot )} ( {\mathcal {D}}_{u} )} \le 2 | {\mathcal {D}}_{u} |^{\frac{1}{p( \zeta )} - \frac{1}{q( \zeta )}} | w |_{L^{q( \cdot )} ( {\mathcal {D}}_{u})} \,\,\, \forall \zeta \in \varOmega . \end{aligned}$$

  It follows that

  $$\begin{aligned} | u |_{L^{p( \cdot )} ( {\mathcal {D}}_{u} )} \le | v |_{L^{p( \cdot )} ( {\mathcal {D}}_{u} )} + | w |_{L^{p( \cdot )} ( \varOmega )}&\le \bigg ( 1 + 2 | {\mathcal {D}}_{u} |^{\frac{1}{p( \zeta )} - \frac{1}{q( \zeta )}} \bigg ) \, | u |_{L^{p( \cdot )} ( {\mathcal {D}}_{u} ) + L^{q( \cdot )} ( {\mathcal {D}}_{u} )} \\ {}&\le \bigg ( 1 + 2 | {\mathcal {D}}_{u} |^{\frac{1}{p( \zeta )} - \frac{1}{q( \zeta )}} \bigg ) \, | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega )}, \end{aligned}$$

  without loss of generality, we may assume that u is nonnegative, such that

  $$\begin{aligned} | u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega )} = | v |_{L^{p( \cdot )} ( \varOmega )} + | w |_{L^{q( \cdot )} ( \varOmega )}. \end{aligned}$$

  From the above definition, it follows that, $0 \le w \le 1$ on ${\mathcal {D}}_{u}^{c}.$ We denote $\sigma _{p} = | w |_{L^{p( \cdot )} ( {\mathcal {D}}^{c}_{u})}$ and $\sigma _{q} = | w |_{L^{q( \cdot )} ( {\mathcal {D}}^{c}_{u})}.$ According to the fact that $| w | \le 1$ on ${\mathcal {D}}_{u}^{c},$ we get

  $$\begin{aligned} 1 = \int _{{\mathcal {D}}_{u}^{c}} \bigg | \frac{w}{\sigma _{p}} \bigg |^{p( x )} \,\, dx \ge \frac{\sigma _{q}^{q( \zeta )}}{\sigma _{p}^{p( \zeta )}} \int _{{\mathcal {D}}_{u}^{c}} \bigg | \frac{w}{\sigma _{q}} \bigg |^{q( x )} \,\, dx = \frac{\sigma _{q}^{q( \zeta )}}{\sigma _{p}^{p( \zeta )}} \,\,\, \forall \zeta \in \varOmega . \end{aligned}$$

  Thus, $\sigma _{p} \ge \sigma _{q}^{\frac{q( \zeta )}{p( \zeta )}}.$ Similarly,

  $$\begin{aligned} | u |_{L^{p( \cdot )} ( {\mathcal {D}}_{u}^{c} ) + L^{q( \cdot )} ( {\mathcal {D}}_{u}^{c})} = | v |_{L^{p( \cdot )} ( {\mathcal {D}}_{u}^{c})} + | w |_{L^{q( \cdot )} ( {\mathcal {D}}_{u}^{c} )}&\ge | v |_{L^{q( \cdot )} ( {\mathcal {D}}_{u}^{c} )}^{\frac{q( \zeta )}{p( \zeta )}} + | w |_{L^{q( \cdot )} ( {\mathcal {D}}_{u}^{c} )}\\&\ge c \min \, \left\{ | u |_{L^{q( \cdot } ({\mathcal {D}}_{u}^{c} )}, \, | u |_{L^{q( \cdot )} ( {\mathcal {D}}_{u}^{c} )}^{\frac{q( \zeta )}{p( \zeta )}} \,\right\} \end{aligned}$$

  According to the previous results, we obtain the result.

Proposition 2

[29] Assume that assumptions $({\mathcal {A}}_{1} )$-(iv) are true. Then $( L^{p'( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega ), \, | \cdot |_{L^{p'( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega )} )$ is a reflexive Banach space.

Now, we consider

$$\begin{aligned} X( \varOmega ) = \{ u \in L^{\alpha ( \cdot )} ( \varOmega )/\, \nabla u \in ( L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega ) )^{N} \,\} \end{aligned}$$

with the norm

$$\begin{aligned} || u ||_{\varOmega } = || u ||_{L^{\alpha ( \cdot )} ( \varOmega )} + | \nabla u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega )}. \end{aligned}$$

Proposition 3

[29] Under assumptions $({\mathcal {A}}_{1} )$-(iv). $( X( \varOmega ), \, || u ||_{\varOmega } )$ is a reflexive and a Banach space.

Theorem 1

[29] Suppose that hypotheses $({\mathcal {A}}_{1} )$-(iv), $1 \ll p^{*} ( \cdot ) \, \frac{q'( \cdot )}{p'( \cdot )}$ hold, $\alpha$ satisfies $1 \ll \alpha ( \cdot ) \ll p^{*} ( \cdot ) \, \frac{N - 1}{N}$ and $1 \ll \alpha ( \cdot ) \le p^{*} \, \frac{q' ( \cdot )}{p'( \cdot )}.$ Then, the space $X( \varOmega )$ is continuously embedded into $L^{p^{*} ( \cdot )} ( \varOmega ).$

Proposition 4

Assume that hypotheses $({\mathcal {A}}_{1} )$-(iv), $1 \ll p^{*} ( \cdot ) \, \frac{q'( \cdot )}{p'( \cdot )}$ hold, $\alpha$ satisfies $1 \ll \alpha ( \cdot ) \ll p^{*} ( \cdot ) \, \frac{N - 1}{N}$ and $1 \ll \alpha ( \cdot ) \le p^{*} \, \frac{q' ( \cdot )}{p'( \cdot )}.$ The following properties are true:

• (i) For any $u \in X( \varOmega ), \, \psi _{n} u \rightarrow u \,\, \text{ in } \,\, X( \varOmega ).$

• (ii) For any $u \in X( \varOmega ),$ we have $u_{\epsilon } = u *j_{\epsilon } \rightarrow u$ in $X( \varOmega )$ (where $j_{\epsilon } ( x ) = \epsilon ^{-N} \, j \bigg ( \frac{x}{\epsilon } \bigg )$ and $j : \varOmega \rightarrow {\mathbb {R}}^{+}$ is in $C_{c}^{\infty } ( \varOmega ),$ a function inducing a probability measure.

• (iii) For any $u \in X( \varOmega ),$ there exists a sequence $\{ u_{n} \} \subset C_{c}^{\infty } ( \varOmega )$ such that $u_{n} \rightarrow u$ in $X( \varOmega ).$

Proof

• (i) See Theorem 3.12 in [29].

• (ii) Using the mollifiers method, we get that $u_{\epsilon } \rightarrow u$ in $L^{\alpha ( \cdot )} ( \varOmega )$ as $\epsilon \rightarrow 0.$ Moreover, if $\nabla u = a + b,$ with $a \in ( L^{p( \cdot )} ( \varOmega ) )^{N}$ and $b \in ( L^{q( \cdot )} ( \varOmega ) )^{N},$ we have $\nabla u_{\epsilon } = \nabla u *j_{\epsilon } = a *j_{\epsilon } + b *j_{\epsilon }$ with $a *j_{\epsilon } \in ( L^{p( \cdot )} ( \varOmega ) )^{N}$ and $b *j_{\epsilon } \in ( L^{q( \cdot )} ( \varOmega ) )^{N}.$ Then

  $$\begin{aligned} | \nabla u_{\epsilon } - \nabla u |_{L^{p( \cdot )} ( \varOmega ) + L^{q( \cdot )} ( \varOmega )} \le | a *j_{\epsilon } - a |_{L^{p( \cdot )} ( \varOmega )} + | b *j_{\epsilon } - b |_{L^{q( \cdot )} ( \varOmega )} \rightarrow 0. \end{aligned}$$

  Hence, $u_{\epsilon } \longrightarrow u$ in $X( \varOmega ).$

• (iii) We can easily conclude the proof of this step, by using the above results (i) and (ii).

3 Properties of functionals and operators

In this section, we begin by presenting some results that can be concluded from the previous assumptions in the Sect. 1.

$\bullet \,$ The conditions $( {\mathcal {A}}_{1} ),$ (i) and (ii) imply that

$$\begin{aligned} {\mathcal {A}} ( x, \zeta ) \le A( x, \zeta )\,.\, \zeta \,\,\, \text{ for } \text{ all } \,\,\, ( x, \zeta ) \in \varOmega \times \varOmega . \end{aligned}$$

(4)

$\bullet \,$ By $( {\mathcal {A}}_{1} ),$ (i) and (iii), we get

$$\begin{aligned} {\mathcal {A}} ( x, \zeta ) = \int _{0}^{1} \frac{\text{ d }}{\text{ d } \theta } {\mathcal {A}} ( x, \theta \zeta ) \,\, \text{ d } \theta = \int _{0}^{1} \frac{1}{\theta } A( x, \theta \zeta ) \,.\, \theta \zeta \,\, \text{ d } \theta \ge {\left\{ \begin{array}{ll} c_{1} \, | \zeta |^{p( x )};& {\text {if}} \quad | \zeta | > 1 , \\ c_{1} \, | \zeta |^{q( x )} ;& {\text {if}} \quad | \zeta | \le 1 , \end{array}\right. } \end{aligned}$$

(5)

$\bullet \,$ According to (iii), (4) and (5) we obtain that

$$\begin{aligned} {\left\{ \begin{array}{ll} c_{1} \, | \zeta |^{p( x )} ;& {\text {if}} \quad | \zeta |> 1 , \\ c_{1} \, | \zeta |^{q( x )} ; &{\text {if}} \quad | \zeta | \le 1 , \end{array}\right. } \le {\mathcal {A}} ( x, \zeta ) \le A( x, \zeta ) \,.\,\zeta \le {\left\{ \begin{array}{ll} c_{2} \, | \zeta |^{p( x )} ;& {\text {if}} \quad | \zeta | > 1 , \\ c_{2} \, | \zeta |^{q( x )} ;& {\text {if}} \quad | \zeta | \le 1 , \end{array}\right. } \end{aligned}$$

(6)

For all $( x, \zeta ) \in \varOmega \times \varOmega .$

By (g) of Proposition 1, we deduce that ${\mathcal {A}} ( x, \nabla u )$ is integrable on $\varOmega$ for all $u \in X.$ Thus, $\int _{\varOmega } {\mathcal {A}} ( x, \nabla u ) \,\, dx$ is well defined. For $u \in X,$ it follows by (6) that

$$\begin{aligned} \int _{\varOmega }&A( x, \nabla u ) \,.\, \nabla u \,\, dx + \int _{\varOmega } | u |^{\alpha ( x )} \,\, dx \nonumber \\&\ge c_{1} \, \bigg ( \int _{\varOmega \cap {\mathcal {D}}_{\nabla u}} | \nabla u |^{p( x )} \,\, dx + \int _{\varOmega \cap {\mathcal {D}}^{c}_{\nabla u}} | \nabla u |^{q( x )} \,\, dx + \int _{\varOmega } | u |^{\alpha ( x )} \,\, dx \bigg ), \end{aligned}$$

(7)

and

$$\begin{aligned} \int _{\varOmega }&A( x, \nabla u ) \,.\, \nabla u \,\, dx + \int _{\varOmega } | u |^{\alpha ( x )} \,\, dx \nonumber \\&\le c_{2} \, \bigg ( \int _{\varOmega \cap {\mathcal {D}}_{\nabla u}} | \nabla u |^{p( x )} \,\, dx + \int _{\varOmega \cap {\mathcal {D}}^{c}_{\nabla u}} | \nabla u |^{q( x )} \,\, dx + \int _{\varOmega } | u |^{\alpha ( x )} \,\, dx \bigg ), \end{aligned}$$

where $c_{1}$ and $c_{2}$ are positive constants.

Similarly, using (6), we get for all $u \in X$

$$\begin{aligned} \int _{\varOmega }&{\mathcal {A}} ( x, \nabla u ) \,\, dx + \int _{\varOmega } \frac{1}{\alpha ( x )} \, | u |^{\alpha ( x )} \,\, dx \nonumber \\&\ge c_{1} \, \bigg ( \int _{\varOmega \cap {\mathcal {D}}_{\nabla u}} | \nabla u |^{p( x )} \,\, dx + \int _{\varOmega \cap {\mathcal {D}}^{c}_{\nabla u}} | \nabla u |^{q( x )} \,\, dx + \int _{\varOmega } \frac{1}{\alpha ( x )} | u |^{\alpha ( x )} \,\, dx \bigg ), \end{aligned}$$

(8)

and

$$\begin{aligned} \int _{\varOmega }&{\mathcal {A}} ( x, \nabla u ) \,\, dx + \int _{\varOmega } \frac{1}{\alpha ( x )} \, | u |^{\alpha ( x )} \,\, dx \\&\le c_{2} \, \bigg ( \int _{\varOmega \cap {\mathcal {D}}_{\nabla u}} | \nabla u |^{p( x )} \,\, dx + \int _{\varOmega \cap {\mathcal {D}}^{c}_{\nabla u}} | \nabla u |^{q( x )} \,\, dx + \int _{\varOmega } \frac{1}{\alpha ( x )} | u |^{\alpha ( x )} \,\, dx \bigg ), \end{aligned}$$

where,

$$\begin{aligned} {\mathcal {D}}_{\nabla u} = \{ x \in \varOmega \, /\, | \nabla u( x ) | > 1 \} \,\,\, \text{ and } \,\,\, {\mathcal {D}}^{c}_{\nabla u} = \{ x \in \varOmega \, / \, |\nabla u( x ) | \le 1 \}. \end{aligned}$$

Let K be a subset of $X( \varOmega )$ defined by

$$\begin{aligned} K = \{ u \in X( \varOmega ) / u( x ) \le \varPhi ( x ) \, \text{ for } \text{ a.a }\,\, x \in \varOmega \} \end{aligned}$$

(9)

Remark 1

• (a) The set K is nonempty, closed and convex subset of $X( \varOmega )$.

• (b) From assumption (vi) we see that $0 \in K.$

Definition 2

• (a) We say that $u \in K$ is a weak solution of problem $( {\mathcal {P}} )$ if

  $$\begin{aligned} \int _{\varOmega } A( x, \nabla u ) \,.\, \nabla ( v - u ) \,\, dx&+ \int _{\varOmega } | u |^{\alpha ( x ) - 2} \, u \,.\, ( v - u ) \,\, dx \\&= \int _{\varOmega } \lambda \, b( x ) \, | u |^{\beta ( x ) - 2} \, u \, ( v - u ) \,\, dx, \end{aligned}$$

  for all $v \in K,$ where K is given by (4).

• (b) We say that $u \in X$ is a weak solution of problem $( {\mathcal {P}} )$ if for all $\varphi \in X,$ we have

  $$\begin{aligned}&\int _{\varOmega } A( x, \nabla u ) \,.\, \nabla \varphi ( x ) \,\, dx + \frac{1}{\rho _{n}} \int _{\varOmega } ( u( x ) - \varPhi ( x ) )^{+} \,.\, \varphi ( x ) \,\, dx \\ {}&+ \int _{\varOmega } | u |^{\alpha ( x ) - 2}\, u\,.\, \varphi ( x ) \,\, dx = \int _{\varOmega } \lambda \, b( x ) \, | u |^{\beta ( x ) - 2} \, u \,.\, \varphi ( x ) \,\, dx, \end{aligned}$$

  where $\{\rho _{n} \}$ is a sequence with $\rho _{n} > 0$ for each $n \in {\mathbb {N}}$ such that $\rho _{n} \longrightarrow 0$ when $n \rightarrow \infty .$

It is easy to prove the following lemma.

Lemma 1

If hypotheses (vi) holds, then the function $I : X \longrightarrow X^{*}$ given by

$$\begin{aligned} \langle I u, \, \varphi \, \rangle _{X} = \int _{\varOmega } ( u( x ) - \varPhi ( x ) )^{+} \,.\, \varphi ( x ) \,\, dx \qquad \text{ for } \text{ all } \,\, u, \varphi \in X, \end{aligned}$$

is bounded, demi-continuous and monotone, where $\langle \cdot , \, \cdot \rangle _{X}$ denotes the duality pairing between X and its dual space $X^{*}.$

Proof

From (vi), we deduce that the function $\varPhi$ is nonnegative. Next, according to the Proposition 1 we get that the function I is bounded, monotone and for the demi-continuous of the function I, we consider $\{u_{n}\}_{n \in {\mathbb {N}}^{*}}$ a bounded sequence in $\varOmega$ such as $u_{n} \rightarrow u$ for all $u \in \varOmega$, we get that $I\,u_{n} \rightharpoonup I\, u$ in $X( \varOmega ).$

Theorem 2

Assume that $1 \ll \beta (\cdot ) \le \alpha (\cdot ) \le p(\cdot ) \ll q(\cdot ) \ll \text{ min } \{ N, p^{*} (\cdot ) \}, \, 1 \ll \alpha ( \cdot ) \ll p^{*} ( \cdot ) \, \frac{q'( \cdot )}{p'( \cdot )}, \, \lambda$ is small enough, and hypotheses $( {\mathcal {A}}_{1} ) - ( {\mathcal {A}}_{2} ),$ (vi), (vii) and (ix) hold. Then, the problem $( {\mathcal {P}} )$ possesses a weak solution.

Proof

It follows that solutions of $( {\mathcal {P}} )$ correspond to the critical points of the Euler-Lagrange energy functional $J : X \longrightarrow {\mathbb {R}}$

$$\begin{aligned} J( u ) =&\int _{\varOmega } {\mathcal {A}} ( x, \nabla u ) \,\, dx + \int _{\varOmega } \frac{1}{\alpha ( x )} \, | u |^{\alpha ( x )} \,\, dx - \frac{1}{\rho _{n}} \int _{\varOmega } ( u ( x ) - \varPhi ( x ) )^{+} \,\, dx \\&- \int _{\varOmega } \lambda \, \frac{b( x )}{\beta ( x )} \, | u |^{\beta ( x )} \,\, dx. \end{aligned}$$

Lemma 2

Assume that $1 \ll \beta (\cdot ) \le \alpha (\cdot ) \le p(\cdot ) \ll q(\cdot ) \ll \text{ min } \{ N, p^{*} (\cdot ) \}, \, 1 \ll \alpha ( \cdot ) \ll p^{*} ( \cdot ) \, \frac{q'( \cdot )}{p'( \cdot )}, \, \lambda$ is small enough, and hypotheses $( {\mathcal {A}}_{1} ) - ( {\mathcal {A}}_{2} ),$ (vi), (vii) and (ix) hold. The functional J satisfies mountain pass geometry in the sense that:

(1):

$J( 0 ) = 0.$

(2):

There exists $\gamma , \, \delta > 0$ such that $J( u ) \ge \delta$ if $|| u ||_{X} > \gamma$.

(3):

There exists $u, \, || u ||_{X} > \gamma$ such that $J( u ) \le 0.$

Proof

• (1) According to (ii) and the nonnegativity of $\varPhi ,$ we deduce that $J( 0 ) = 0.$

• (2) According to Lemma 1, (vi),  (vii),  (ix) and the monotonicity of the function $s \longmapsto s^{+},$ we get

  $$\begin{aligned} J( u )&= \int _{\varOmega } {\mathcal {A}} ( x, \nabla u ) \,\, dx + \int _{\varOmega } \frac{1}{\alpha ( x )} | u |^{\alpha ( x )} \,\, dx - \frac{1}{\rho _{n}} \int _{\varOmega } ( u( x ) - \varPhi ( x ) )^{+} \,\, dx \\&\quad - \int _{\varOmega } \lambda \, \frac{b( x )}{\beta ( x )} | u |^{\beta ( x )} \,\, dx \\&\ge c_{1} \bigg ( \int _{\varOmega \cap \Lambda _{\nabla u}} | \nabla u |^{p( x )} \,\, dx + \int _{\varOmega \cap \Lambda ^{c}_{\nabla u}} | \nabla u |^{q( x )} \,\, dx + \int _{\varOmega } \frac{1}{\alpha ( x )} | u |^{\alpha ( x )} \,\, dx \bigg ) \\&\quad - \frac{1}{\rho _{n}} \int _{\varOmega } ( u( x ) - \varPhi ( x ) )^{+} \,\, dx - \int _{\varOmega } \lambda \, \frac{b( x )}{\beta ( x )} | u |^{\beta ( x )} \,\, dx\\&\ge c_{1} \bigg ( \frac{1}{p^{+}} + \frac{1}{q^{+}} + \frac{1}{\alpha ^{-}} \, \bigg ) || u ||^{\alpha ^{-}}_{X} - \lambda \, c_{2} \, || u ||_{X}^{\beta ^{+}} \longrightarrow + \infty , \, \text{ when } \, || u ||_{X} \rightarrow + \infty . \end{aligned}$$

  Then, $\exists \eta > 0$ be small enough. Which for that, we have

  $$\begin{aligned} J( u ) \ge \delta > 0 \text{ for } \text{ all } \,\, || u ||_{X} = \eta . \end{aligned}$$

• (3) Let K be a real fixed, choosing $\varUpsilon _{k}$ a k-dimensional linear subspace of X such that $\varUpsilon _{k} \subset C_{0}^{\infty } ( B_{R} ),$ which the norms on $\varUpsilon _{k}$ are equivalent. Then, for any $\delta _{0} > 0$ given, there exists $\sigma _{k} \in (0, \, 1 )$ such that $u \in \varUpsilon _{k}$ with $|| u || \le \sigma _{k}$ implies $| u |_{L^{\infty }} \le \delta _{0}.$ Consider the following set:

  $$\begin{aligned} F_{\sigma _{k}}^{( k )} = \{ u \in \varUpsilon _{k} : \, || u || = \sigma _{k} \,\}. \end{aligned}$$

  For $u \in F_{\sigma _{k}}^{( k )}$ and $0< t < 1,$ we get

  $$\begin{aligned} J( tu )&= \int _{\varOmega } {\mathcal {A}} ( x, \nabla tu ) \,\, dx + \int _{\varOmega } \frac{1}{\alpha ( x )} \, | tu |^{\alpha ( x )} \,\, dx - \frac{1}{\rho _{n}} \int _{\varOmega } ( tu ( x ) - \varPhi ( x ) )^{+} \,\, dx \\&\quad - \int _{\varOmega } \lambda \, \frac{b( x )}{\beta ( x )} \, | tu |^{\beta ( x )} \,\, dx \\&\le c_{1} \frac{t^{\alpha ^{-}}}{\alpha ^{-}} \, \sigma _{k}^{\alpha ^{-}} - \frac{1}{\rho _{n}}\,t\, \sigma _{k} - t^{\beta ^{+}} \, \theta _{k}, \end{aligned}$$

  because, by the compactness of the set $F_{\sigma _{k}}^{(k)}, \, \exists \, \theta _{k} > 0$ such that

  $$\begin{aligned} \int _{\varOmega } \lambda \, \frac{b( x )}{\beta ( x )} | tu |^{\beta ( x )} \,\, dx \ge \theta _{k} \,\,\,\, \text{ for } \text{ all } \,\,\, u \in F_{\sigma _{k}}^{(k)}. \end{aligned}$$

  Hence, since $1 \ll \beta ^{+} < \alpha ^{-},$ we find $t_{k} \in ( 0,\, 1)$ and $\epsilon _{k} > 0$ such that

  $$\begin{aligned} J( t_{k} u ) \le - \epsilon _{k} < 0 \qquad \text{ for } \text{ all } \,\, u \in F_{\sigma _{k}}^{(k)}. \end{aligned}$$

  Then,

  $$\begin{aligned} J( u ) \le - \epsilon _{k} < 0 \quad \text{ for } \text{ all } \,\,\, u \in F_{t_{k} \,\sigma _{k}}^{(k)}. \end{aligned}$$

  which completes the proof of the lemma.

Lemma 3

Assume that $1 \ll \beta (\cdot ) \le \alpha (\cdot ) \le p(\cdot ) \ll q(\cdot ) \ll \text{ min } \{ N, p^{*} (\cdot ) \}, \, 1 \ll \alpha ( \cdot ) \ll p^{*} ( \cdot ) \, \frac{q'( \cdot )}{p'( \cdot )}, \, \lambda$ is small enough, and hypotheses $( {\mathcal {A}}_{1} ) - ( {\mathcal {A}}_{2} ),$ (vi), (vii) and (ix) hold. The functional J satisfies Palais–Smale condition.

Proof

Let $\{ u_{n} \}$ be a Palais–Smale condition sequence, such that the associated sequence of real numbers $\{ J( u_{n} ) \}$ is bounded, and $J'( u_{n} ) \longrightarrow 0$ in $X^{*}.$ For that we will demonstrate the $( u_{n} )$ is bounded in X, and we will argue it by contradiction.

We suppose that $|| u_{n} ||_{X} \rightarrow \infty$ when $n \rightarrow \infty .$ Then, we have

$$\begin{aligned} J( u_{n} ) - \frac{1}{\mu } \langle J'( u_{n} ), \, u_{n} \, \rangle&= \int _{\varOmega } {\mathcal {A}} ( x, \nabla u_{n} ) \,\, dx + \int _{\varOmega } \frac{1}{\alpha ( x )} \, | u_{n} |^{\alpha ( x )} \,\, dx \nonumber \\&\quad - \frac{1}{\rho _{n}} \int _{\varOmega } ( u_{n} ( x ) - \varPhi ( x ) )^{+} \,\, dx - \int _{\varOmega } \lambda \, \frac{b( x )}{\beta ( x )} \, | u_{n} |^{\beta ( x )} \,\, dx \nonumber \\&\quad - \frac{1}{\mu } \int _{\varOmega } A( x, \nabla u_{n} ) \,.\, \nabla u_{n} \,\, dx - \frac{1}{\mu } \int _{\varOmega } | u_{n} |^{\alpha ( x )} \,\,dx \nonumber \\&\quad + \frac{1}{\mu } \, \frac{1}{\rho _{n}} \int _{\varOmega } ( u_{n} ( x ) - \varPhi ( x ))^{+} \,\, dx + \frac{1}{\mu } \int _{\varOmega } \lambda \, b( x ) \, | u_{n} |^{\beta ( x )} \,\, dx \nonumber \\&\ge c_{1} \int _{\varOmega } {\mathcal {A}} ( x, \nabla u_{n} ) + | u_{n} |^{\alpha ( x )} \,\, dx \nonumber \\&\quad + \bigg ( \frac{1}{\mu } - 1 \bigg ) \, \frac{1}{\rho _{n}} \int _{\varOmega } ( u_{n} ( x ) - \varPhi ( x ) )^{+} \,\, dx \nonumber \\&\quad + \bigg ( \frac{1}{\mu } - \frac{1}{\beta ^{+}} \bigg ) \int _{\varOmega } \lambda \, b( x ) | u_{n} |^{\beta ( x )} \,\, dx \nonumber \\&\ge c_{1} \, || u_{n} ||^{\alpha ^{-}}_{X} - c_{2} \, || u_{n} ||_{X} - c_{3} \, || u_{n} ||_{X}^{\beta ^{+}} - c_{4}, \end{aligned}$$

(10)

where, $c_{1}, c_{2}, c_{3}$ and $c_{4}$ are positive constants.

Dividing both sides of (10) by $|| u_{n} ||_{X}$ and passing to the limit $n \rightarrow + \infty$ with the fact that $\alpha ^{-}> \beta ^{+} > 1,$ we get $0 \ge \infty .$ Which is a contradiction.

Hence, the functional J satisfies the Palais–Smale condition.

4 Conclusion

According to the Lemmas 2 and 3, we conclude that the problem $( {\mathcal {P}} )$ possesses a weak solution.