Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces

Abstract

In many physical models, internal energy will run out without external energy sources. Therefore, finding optimal energy sources and studying their behavior are essential issues. In this article, we study the following variational problem: ${\mathcal{G}}_{ϵ}^{\ast }=sup\left\{{\int }_{\Omega }\frac{G\left(u\right)}{{ϵ}^{q\left(x\right)}}dx:{\parallel \nabla u\parallel }_{L^{p(.)}}\le ϵ,u=0 \mathrm{on} \partial \Omega \right\},$ with the help of $\Gamma$-convergence, where $G:\mathbb{R}\to \mathbb{R}$ is upper semicontinuous, non zero in the $L^1$ sense, $0\le G\left(u\right)\le {c|u|}^{q(.)}$, $\Omega$ is a bounded open subset of ${\mathbb{R}}^n$, $n\ge 3$, $1<p(.)<n$, and $p(.)\le q(.)\le p^{\ast }(.)$. For special choices of G, we can study Bernoulli’s free-boundary and plasma problems in variable exponent Lebesgue spaces.

1. Introduction

The aim of this paper is to study the asymptotic behavior of low energy extremals $\left\{u_{ϵ}\right\}\subseteq W_0^{1,p(.)}(\Omega )$ for which the following supremum is attained:

$${\mathcal{G}}_{ϵ}^{\ast }=sup\left\{{\int }_{\Omega }\frac{G\left(u\right)}{{ϵ}^{q\left(x\right)}}dx:u\in W_0^{1,p(.)}(\Omega ),{\parallel \nabla u\parallel }_{L^{p(.)}}\le ϵ,u=0 \mathrm{on} \partial \Omega \right\},$$

(1)

with $\Gamma$-convergence, when $ϵ\to 0$. Here, $\Omega$ is a bounded open subset of ${\mathbb{R}}^n$, $n\ge 3$, and $G:\mathbb{R}\to \mathbb{R}$ is a upper semicontinuous map, non zero in the $L^1$ sense, and satisfies the following growth condition:

$$0\le G\left(u\right)\le {c|u|}^{q(.)}, \mathrm{for} \mathrm{all} u\in W_0^{1,p(.)}(\Omega ).$$

(2)

Additionally, the exponents $p,q:\overline{\Omega }\to \mathbb{R}$ satisfy the following conditions:

(C1)

$1<p^{-}=infp\le supp=p^{+}<n$ over $\overline{\Omega }$;

(C2)

$p(.)\le q(.)\le p^{\ast }(.)$, where $p^{\ast }\left(x\right)=\frac{np\left(x\right)}{n-p\left(x\right)}$, for $x\in \overline{\Omega }$;

(C3)

p and q are Log-Hölder continuous;

(C4)

The set $\mathcal{C}=\{x\in \overline{\Omega }:p^{\ast }\left(x\right)=q\left(x\right)\}\ne \varnothing$.

We refer to the next section for the details of variable exponent Sobolev space. In our study, due to Condition (C4), the embedding $W^{1,p(.)}(\Omega )\hookrightarrow L^{q(.)}(\Omega )$ is not compact.

For a constant exponent, Problem (1) and its variants were extensively studied; see [1,2,3,4,5,6] and the references therein. For $p=2$, $G\left(s\right)={|s|}^{2^{\ast }}:s\in \mathbb{R}$, Lions [7] used the concentration–compactness principle to prove that maximizing sequence of (1) either converges to a single point of compact; the concentration-compactness principle was also used to prove the existence of a solution for many kinds of elliptic PDEs with critical growth (see [8,9,10,11]). Later on, Flucher and Müller [2] generalized Lions’ concentration–compactness principle to study Problem (1) for a constant exponent and discussed its various applications.

Bonder and Silva [12] and Yongqiang [13] proved the concentration/compactness principle for variable exponent Lebesgue spaces, independently, to deal with the following nonlinear elliptic PDE with critical growth at infinity:

$$-{▵}_{p(.)}v=g(.,v),$$

(3)

where ${▵}_{p\left(x\right)}v:={div(|\nabla v|}^{p\left(x\right)-2}\nabla v)$ is known as a $p\left(x\right)$-Laplacian operator. Critical growth is known as

$$|g(.,s)|\le c(1+|s{|}^{q(.)}).$$

(4)

Many authors have studied models such as electrorheological fluids with variants of the above elliptic PDE with different boundary conditions (Neumann, Dirichlet, and nonlinear); see [14,15,16,17] and references therein.

Recently, Bashir et al. [18] extended the work of Bonder and Silva [12] and proved generalized concentration/compactness principles for variable exponent Lebesgue spaces. In addition, they initiated the work on Problem (1) and proved the existence of low energy extremals. $\Gamma$-convergence is a significant variational convergence to study the asymptotic behavior of extremals of a sequence of functionals; typically, the $\Gamma$-convergence limit retains the desired features of a sequence.

This approach has already been used in similar scenarios. Bonder et al. [19] used $\Gamma$-convergence to determine the subcritical approximation of the best constant associated with critical Sobolev embedding $W^{1,p(.)}(\Omega )\hookrightarrow L^{p^{\ast }(.)}(\Omega )$. Briane et al. [20] studied the asymptotic behavior of the sequence of equi-coercive nonlinear energies defined on vector-valued functions. Amaziane et al. [21] worked with homogenization and minimization problems for a class of variational functionals, using $\Gamma$-convergence in the framework of Sobolev spaces with continuous variable exponents. Carbotti et al. [22] proved the local minimality of half spaces in Carnot groups and provided a lower bound of $\Gamma$-lim inf of the recalled energy in the form of horizontal parameters. Kolditz and Mang [23] studied the numerical solutions of quasi-static phase-field fracture problems based upon the $\Gamma$-convergence of parameters. For a comprehensive study on $\Gamma$-convergence, we refer to [24].

In Section 2, we present some preliminary concepts. Section 4 deals with the proofs of the main results of Theorems 1 and 2. In Section 5, we give some concluding remarks and possible future lines of research.

2. Preliminaries

To work in the framework of variable exponent Lebesgue spaces, we present some preliminary concepts here; for a detailed study, see [25]. Variable exponent Lebesgue space with exponent $p(.)$ is defined as

$$L^{p(.)}(\Omega )=\left\{u\in L_{loc}^1(\Omega ):{\int }_{\Omega }{|u\left(x\right)|}^{p(.)}dx<\infty \right\},$$

with the following norm

$${\parallel u\parallel }_{L^{p(.)}(\Omega )}=inf\left\{\sigma >0:{\int }_{\Omega }{|u\left(x\right)/\sigma |}^{p(.)}dx\le 1\right\}.$$

The variable exponent Sobolev space is defined as

$$W^{1,p(.)}(\Omega )=\left\{u\in W_{loc}^{1,1}(\Omega ):u\in L^{p(.)}(\Omega ),|\nabla u|\in L^{p(.)}\right\},$$

with the norm

$${\parallel u\parallel }_{W^{1,p(.)}(\Omega )}={\parallel u\parallel }_{L^{p(.)}(\Omega )}+{\parallel \nabla u\parallel }_{L^{p(.)}(\Omega )}.$$

The closure of $C_c^{\infty }(\Omega )$ in $W^{1,p(.)}(\Omega )$ is denoted as $W_0^{1,p(.)}(\Omega )$. For $1<infp(.)\le supp(.)<\infty ,$ the spaces $L^{p(.)}(\Omega )$, $W^{1,p(.)}(\Omega )$ and $W_0^{1,p(.)}(\Omega )$ are reflexive and separable Banach spaces.

Proposition 1.

Let p and q be exponents satisfying conditions (C1)–(C4). Then, we have the continuous embedding

$$W^{1,p(.)}(\Omega )\hookrightarrow L^{q(.)}(\Omega ),$$

Moreover, the above embedding is compact if ${inf}_{x\in \overline{\Omega }}(p^{\ast }\left(x\right)-q\left(x\right))>0$.

Proposition 2

(Poincaré inequality). For all u in $W_0^{1,p(.)}(\Omega )$, we have

$${||u||}_{L^{p(.)}(\Omega )}\le {K||\nabla u||}_{L^{p(.)}(\Omega )}.$$

Recall the notations:

$$q^{+}=\underset{\Omega }{sup}q\left(x\right),q^{-}=\underset{\Omega }{inf}q\left(x\right).$$

Let $\rho \left(u\right):={\int }_{\Omega }{|u\left(x\right)|}^{p\left(x\right)}dx$, then the following proposition is useful to deal with the variable exponent norm.

Proposition 3

([26]). For $u\in L^{p\left(x\right)}(\Omega )$ and the sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subseteq L^{p\left(x\right)}(\Omega )$,

$$u\ne 0\Rightarrow \left({\parallel u\parallel }_{L^{p\left(x\right)}(\Omega )}=\lambda \iff \rho \left(\frac{u}{\lambda }\right)=1\right).$$

(5)

$${\parallel u\parallel }_{L^{p\left(x\right)}(\Omega )}<1(=1;>1)\iff \rho \left(u\right)<1(=1;>1).$$

(6)

$${\parallel u\parallel }_{L^{p\left(x\right)}(\Omega )}>1\Rightarrow {\parallel u\parallel }_{L^{p\left(x\right)}(\Omega )}^{p^{-}}\le \rho \left(u\right)\le {\parallel u\parallel }_{L^{p\left(x\right)}(\Omega )}^{p^{+}}.$$

(7)

$${\parallel u\parallel }_{L^{p\left(x\right)}(\Omega )}<1\Rightarrow {\parallel u\parallel }_{L^{p\left(x\right)}(\Omega )}^{p^{+}}\le \rho \left(u\right)\le {\parallel u\parallel }_{L^{p\left(x\right)}(\Omega )}^{p^{-}}.$$

(8)

$$\underset{n\to \infty }{lim}{\parallel u_n\parallel }_{L^{p\left(x\right)}(\Omega )}=0\iff \underset{n\to \infty }{lim}\rho \left(u_n\right)=0.$$

(9)

$$\underset{n\to \infty }{lim}{\parallel u_n\parallel }_{L^{p\left(x\right)}(\Omega )}=\infty \iff \underset{n\to \infty }{lim}\rho \left(u_n\right)=\infty .$$

(10)

Remark 1.

The norms ${||\nabla u||}_{L^{p(.)}(\Omega )}$ and ${||u||}_{W^{1,p(.)}(\Omega )}$ are equivalent for the Sobolev space $W_0^{1,p(.)}(\Omega ).$

In the end, we present two lemmas from [18] for use in a later section.

Lemma 1.

Consider ${\mathcal{G}}^{\ast }(p(.),q(.),{\mathbb{R}}^N)$ and $ϵ<1$. Then,

(a) 

${\mathcal{G}}_{ϵ}^{\ast }(p(.),q(.),\Omega )\le {\mathcal{G}}^{\ast }$;

(b) 

${\mathcal{G}}^{\ast }={lim}_{ϵ\to 0}{\mathcal{G}}_{ϵ}^{\ast }(p(.),q(.),\Omega ).$

Lemma 2.

Let v in $W_0^{1,p(.)}(\Omega )$ with ${\parallel \nabla v\parallel }_{L^{p(.)}(\Omega )}\le ϵ$. Take $x_0\in \Omega$, $u=v/ϵ$, $\delta >0$, $r<R$ satisfying $\frac{r}{R}\le k\left(\delta \right)$. Let

$$p_R^{+}=\underset{x\in B_R\left(x_0\right)}{sup}p\left(x\right), p_R^{-}=\underset{x\in B_R\left(x_0\right)}{inf}p\left(x\right),$$

$$q_R^{+}=\underset{x\in B_R\left(x_0\right)}{sup}q\left(x\right), q_R^{-}=\underset{x\in B_R\left(x_0\right)}{inf}q\left(x\right).$$

For G satisfying the growth condition (2), the following inequality holds:

$$\begin{array}{cc}\hfill {\int }_{B_r\left(x_0\right)}\frac{G\left(v\right)}{{ϵ}^{q\left(x\right)}}dx& \le G_{ϵ}^{\ast }max\left\{{\left({\int }_{B_R\left(x_0\right)}{|\nabla u|}^{p\left(x\right)}dx+\delta max\left\{{\parallel \nabla u\parallel }_{L^{p(.)}}^{p_R^{+}},{\parallel \nabla u\parallel }_{L^{p(.)}}^{p_R^{-}}\right\}\right)}^{\frac{q_R^{+}}{p_R^{-}}},\right.\hfill \\ \hfill & \left.{\left({\int }_{B_R\left(x_0\right)}{|\nabla u|}^{p\left(x\right)}dx+\delta max\left\{{\parallel \nabla u\parallel }_{L^{p(.)}}^{p_R^{+}},{\parallel \nabla u\parallel }_{L^{p(.)}}^{p_R^{-}}\right\}\right)}^{\frac{q_R^{-}}{p_R^{+}}}\right\},\hfill \end{array}$$

(11)

3. Proposed Theorems

First, we fix some notions to present the main result:

• Best Sobolev constant,

  $$\mathcal{S}(p(.),q(.),\Omega )=sup\left\{{\int }_{\Omega }{|u|}^{q(.)}dx:u\in W_0^{1,p(.)}(\Omega ),{\parallel \nabla u\parallel }_{L^{p(.)}(\Omega )}\le 1\right\},$$

  (12)
• Generalized Sobolev constant,

  $${\mathcal{G}}^{\ast }(p(.),q(.),\Omega ):=sup\left\{{\int }_{\Omega }G\left(u\right)dx:u\in W_0^{1,p(.)}(\Omega ) \mathrm{and} {\parallel \nabla u\parallel }_{L^{p(.)}(\Omega )}\le 1\right\},$$

  (13)
•  

  $$\begin{array}{cc}\hfill G_0^{+} :=\underset{s\to 0}{lim\; sup}\frac{G\left(s\right)}{{|s|}^{q^{+}}},& G_0^{-} :=\underset{s\to 0}{lim\; inf}\frac{G\left(s\right)}{{|s|}^{q^{+}}},\hfill \\ \hfill G_{\infty }^{+} :=\underset{|s|\to \infty }{lim\; sup}\frac{G\left(s\right)}{{|s|}^{q^{-}}},& G_{\infty }^{-} :=\underset{|s|\to \infty }{lim\; inf}\frac{G\left(s\right)}{{|s|}^{q^{-}}},\hfill \end{array}$$
• Moreover, denote $G_0:=G_0^{+}=G_0^{-}$ and $G_{\infty }:=G_{\infty }^{+}=G_{\infty }^{-}$. In the case of equality, we assume that the following function exists:

  $$T\left(x\right):=\underset{s\to 0}{lim}\frac{G\left(s\right)}{s^{q\left(x\right)}}:x\in \Omega .$$

To apply $\Gamma$-convergence, we consider the following functional:

$${\mathcal{H}}_{ϵ}\left(u\right):={\int }_{\Omega }\frac{G\left(ϵu\right)}{{ϵ}^{q\left(x\right)}}dx.$$

(14)

Problem (1) is rewritten as

$${\mathcal{G}}_{ϵ}^{\ast }=sup\left\{{\mathcal{H}}_{ϵ}\left(u\right):u\in W_0^{1,p(.)}(\Omega ),{\parallel \nabla u\parallel }_{L^{p(.)}}\le 1\right\}.$$

(15)

However, for any sequence $\left\{u_{ϵ}\right\}$ in $W_0^{1,p(.)}(\Omega )$, any functional defined on $W_0^{1,p(.)}(\Omega )$ alone cannot retain the desired properties of ${\mathcal{H}}_{ϵ}\left(u_{ϵ}\right)$. In this kind of scenario, the idea is to work on the ordered pair $(u_{ϵ},|\nabla u_{ϵ}{|}^{p(.)}dx)$ and $|\nabla u_{ϵ}{|}^{p(.)}dx$ is taken as a measure. Therefore, we consider the following space

$$X(\Omega )=\left\{(u,\mu )\in W_0^{1,p(.)}(\Omega )\times M\left(\overline{\Omega }\right):\mu \left(\overline{\Omega }\right)\le 1,\mu ={|\nabla u|}^{p(.)}dx+\overline{\mu }+\sum _{i\in I}{\mu }_i{\delta }_{x_i}\right\},$$

where $M\left(\overline{\Omega }\right)$ is the set of all finite nonnegative finite Borel measures on $\overline{\Omega }$ and $\overline{\mu }$ is a non-atomic measure in the decomposition of $\mu$. Furthermore, we define the convergence in this space as

$$(u_{ϵ},{\mu }_{ϵ})\stackrel{\mathcal{T}}{\to }(u,\mu )\iff \left\{\begin{array}{cc}{u}_{ϵ}\rightharpoonup u,\hfill & \mathrm{weakly} \mathrm{in} L^{q(.)}(\Omega );\hfill \\ {\mu }_{ϵ}\stackrel{\ast }{\rightharpoonup }\mu ,\hfill & \mathrm{in} \mathrm{the} \mathrm{sense} \mathrm{of} \mathrm{measure} \mathrm{in} M\left(\overline{\Omega }\right).\hfill \end{array}\right.$$

We extend ${\mathcal{H}}_{ϵ}$ to $X(\Omega )$ as follows:

$${\mathcal{H}}_{ϵ}(u,\mu )=\left\{\begin{array}{cc}{\int }_{\Omega }\frac{G\left(ϵu\right)}{{ϵ}^{q\left(x\right)}}dx,\hfill & \mathrm{if} \mu ={|\nabla u|}^{p(.)}dx;\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

In our investigation, we prove the following functional as the $\Gamma$-convergence limit of ${\mathcal{H}}_{ϵ}$

$$\overline{\mathcal{H}}(u,\mu )={\int }_{\Omega }T\left(x\right){|u|}^{q\left(x\right)}dx+{\mathcal{G}}^{\ast }\sum _{i\in I}{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}},$$

(16)

where ${\left\{x_i\right\}}_{i\in I}\subseteq \mathcal{C}$ are the atoms of a limiting measure. Our main result is the following.

Theorem 1.

Let p and q be the exponents satisfying (C1)–(C4). The sequence of functionals $\left\{{\mathcal{H}}_{ϵ}\right\}$ Γ-converges to the functional $\overline{\mathcal{H}}$, i.e., the following two statements hold true for every $(u,\mu )\in X(\Omega )$:

1. 

For every sequence of pairs $\left\{(u_{ϵ},{\mu }_{ϵ})\right\}$ in $X(\Omega )$ with $(u_{ϵ},{\mu }_{ϵ})\stackrel{\mathcal{T}}{\to }(u,\mu )$, we have the inequality

$$\underset{ϵ\to 0}{lim\; sup}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})\le \overline{\mathcal{H}}(u,\mu ),$$

2. 

There exists a sequence of pairs $\left\{(u_{ϵ},{\mu }_{ϵ})\right\}$ in $X(\Omega )$ with $(u_{ϵ},{\mu }_{ϵ})\stackrel{\mathcal{T}}{\to }(u,\mu )$, which satisfies the following inequality:

$$\underset{ϵ\to 0}{lim\; inf}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})\ge \overline{\mathcal{H}}(u,\mu ).$$

Remark 2.

In Theorem 1, what we call Γ-convergence is known generally as ${\Gamma }^{+}$-convergence; see [24]. Γ-convergence is a proper variational convergence to study the convergence of maximizers. More explicitly, if $\left\{u_{ϵ}\right\}$ is the sequence of low energy extremals (maximizers) of ${\mathcal{H}}_{ϵ}$ (i.e., ${\mathcal{H}}_{ϵ}\left(u_{ϵ}\right)={\mathcal{G}}_{ϵ}^{\ast }$ or more generally, ${\mathcal{H}}_{ϵ}\left(u_{ϵ}\right)={\mathcal{G}}_{ϵ}^{\ast }(1+o\left(1\right))$ for $ϵ\to 0^{+}$), $u_{ϵ}\rightharpoonup u$ in $w-L^{q(.)}(\Omega )$ and $|\nabla u_{ϵ}{|}^{p(.)}dx\stackrel{\ast }{\rightharpoonup }\mu$ in $M\left(\overline{\Omega }\right)$, then $(u,\mu )$ is a maximizer of $\overline{\mathcal{H}}$. Furthermore,

$${\mathcal{G}}_{ϵ}^{\ast }\to \underset{X(\Omega )}{max}\overline{\mathcal{H}}.$$

In fact, we prove that $max\overline{\mathcal{H}}={\mathcal{G}}^{\ast }$.

Theorem 2.

Let $\left\{(u_{ϵ},{\mu }_{ϵ})\right\}\subseteq X(\Omega )$, be a maximizing sequence for the functional ${\mathcal{H}}_{ϵ}$ with ${\mu }_{ϵ}={|\nabla u_{ϵ}|}^{p(.)}dx$. Then as $ϵ\to 0$, this sequence concentrates at a single point in the sense that $(u_{ϵ},{\mu }_{ϵ})\stackrel{\mathcal{T}}{\to }(0,{\delta }_x),$ where $x\in \overline{\Omega }.$ Further, for each $x\in \overline{\Omega }$, there exists a maximizing sequence which concentrates at x.

In general, for Problem (1), we can study many intersecting scenarios as applications. For example, with $G\left(s\right)=\left\{\begin{array}{cc}0,\hfill & s<1;\hfill \\ 1,\hfill & s>1.\hfill \end{array}\right.$ we can work on Bernoulli’s free boundary problem as the following volume functional:

$${\mathcal{G}}_{ϵ}^V=sup\left\{{\int }_{\{ϵu\ge 1\}}{ϵ}^{-p^{\ast }\left(x\right)}dx:u\in W_0^{1,p(.)}(\Omega ),{\parallel \nabla u\parallel }_{L^{p(.)}}\le 1\right\}.$$

Equivalently, we can write as

$${\mathcal{G}}_{ϵ}^V=max\left\{{\int }_A{ϵ}^{-p^{\ast }\left(x\right)}dx:A\subset \Omega ,cap\left(A\right)\le ϵ\right\},$$

where $cap\left(A\right)$ is the harmonic capacity of the subset A with respect to $\Omega$, known as follows:

$$cap\left(A\right)=inf\left\{{\parallel \nabla u\parallel }_{L^{p(.)}(\Omega )}:u\in W_0^{1,p(.)}(\Omega ),u\ge 1 \mathrm{a}.\mathrm{e}. \mathrm{on} A\right\}.$$

Bernoulli’s free boundary problem naturally arises in electrostatics, fluid dynamics, optimal insulation, and electrochemistry. We intend to work on such problems in the future.

4. Proofs of Theorems

The first task is to determine the possible $\Gamma$-convergence limit, in this regard: for the generalized concentration/compactness principle ([18], Theorem 3) plays a vital role, but during our work on establishing the $\Gamma$-convergence, we realize that there is a need to refine the generalized concentration/compactness principle and find more sharp bounds. Therefore, we prove a refined version (of [18], Theorem 3) before proving the main result.

Theorem 3.

Let p and q be exponents satisfying (C1)–(C4). Let $\left\{v_{ϵ}\right\}$ be a sequence in $W_0^{1,p(.)}(\Omega )$ with $\parallel \nabla v_{ϵ}{\parallel }_{L^{p(.)}(\Omega )}\le ϵ$. Take $u_{ϵ}:=v_{ϵ}/ϵ$, if the following hold:

• $u_{ϵ}\rightharpoonup u$ weakly in $W_0^{1,p(.)}(\Omega )$,
• $|\nabla u_{ϵ}{|}^{p(.)}dx\stackrel{\ast }{\rightharpoonup }\mu$ in $\mathcal{M}\left(\overline{\Omega }\right)$,
• ${ϵ}^{-q(.)}G\left(v_{ϵ}\right)dx\stackrel{\ast }{\rightharpoonup }\nu$ in $\mathcal{M}\left(\overline{\Omega }\right)$.

Then, for a countable index set I,

$$\mu ={|\nabla u|}^{p(.)}dx+\overline{\mu }+\sum _{i\in I}{\mu }_i{\delta }_{x_i}, \mu \left(\overline{\Omega }\right)\le 1,$$

(17)

$$\nu =h dx+\sum _{i\in I}{\nu }_i{\delta }_{x_i}, \nu \left(\overline{\Omega }\right)\le {\mathcal{G}}^{\ast },$$

(18)

where ${\left\{x_i\right\}}_{i\in I}\subseteq \mathcal{C}$, $\overline{\mu }$ is a positive non-atomic measure in $\mathcal{M}\left(\overline{\Omega }\right)$ and $h\in L^1(\Omega )$. Moreover, the atomic and regular parts satisfy the following generalized Sobolev-type inequalities:

$${\nu }_i\le {\mathcal{G}}^{\ast }{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}},$$

(19)

$$\nu \left(\overline{\Omega }\right)\le {\mathcal{G}}^{\ast }max\left\{\mu {\left(\overline{\Omega }\right)}^{\frac{q^{+}}{p^{-}}},\mu {\left(\overline{\Omega }\right)}^{\frac{q^{-}}{p^{+}}}\right\},$$

(20)

$${\int }_{\Omega }hdx\le {\mathcal{G}}^{\ast }max\left\{{\left({\int }_{\Omega }{|\nabla u|}^{p\left(x\right)}dx+\overline{\eta }\left(\overline{\Omega }\right)\right)}^{\frac{q^{+}}{p^{-}}},{\left({\int }_{\Omega }{|\nabla u|}^{p\left(x\right)}dx+\overline{\eta }\left(\overline{\Omega }\right)\right)}^{\frac{q^{-}}{p^{+}}}\right\},$$

(21)

$$h\le {T|u|}^{q(.)} \mathrm{a}.\mathrm{e}. \mathrm{in} \Omega ,$$

(22)

$${\int }_{\Omega }hdx\le G_0^{+}\mathcal{S}max\left\{{\left({\int }_{\Omega }{|u|}^{p\left(x\right)}dx\right)}^{\frac{q^{+}}{p^{-}}},{\left({\int }_{\Omega }{|u|}^{p\left(x\right)}dx\right)}^{\frac{q^{-}}{p^{+}}}\right\}.$$

(23)

Remark 3.

Note that the refinement is done in Inequalities (19) and (22) in comparison to Ref. [18], Theorem 3. Thus we only need to prove these inequalities.

Proof. 

For $y\in \overline{\Omega }$ by Lemma 2

$$\left[{ϵ}^{-q(.)}G\left(v_{ϵ}\right)dx\right]\left({\overline{B}}_r\left(y\right)\right)\le {\mathcal{G}}_{ϵ}^{\ast }max\left\{{\left(|\nabla u_{ϵ}{|}^{p(.)}dx\left({\overline{B}}_R\left(y\right)\right)+\delta \right)}^{\frac{q_R^{+}}{p_R^{-}}},{\left(|\nabla u_{ϵ}{|}^{p(.)}dx\left({\overline{B}}_R\left(y\right)\right)+\delta \right)}^{\frac{q_R^{-}}{p_R^{+}}}\right\},$$

By Lemma 1, ${\mathcal{G}}_{ϵ}^{\ast }\to {\mathcal{G}}^{\ast }$ when $ϵ\to 0$. Hence,

$$\nu \left(y\right)\le \nu \left({\overline{B}}_r\left(y\right)\right)\le {\mathcal{G}}^{\ast }max\left\{{\left(\mu \left({\overline{B}}_R\left(y\right)\right)+\delta \right)}^{\frac{q_R^{+}}{p_R^{-}}},{\left(\mu \left({\overline{B}}_R\left(y\right)\right)+\delta \right)}^{\frac{q_R^{-}}{p_R^{+}}}\right\},$$

Now, $p_R^{-}$ and $p_R^{+}$ converge to $p\left(y\right)$, whereas $q_R^{-}$ and $q_R^{+}$ converge to $q\left(y\right)$, when $R\to 0$. Hence, taking $r\to 0$, $R\to 0$ and $\delta \to 0$

$$\nu \left(y\right)\le {\mathcal{G}}^{\ast }\mu {\left(y\right)}^{\frac{q\left(y\right)}{p\left(y\right)}}.$$

In particular, for atoms $\left\{x_i\right\}$,

$${\nu }_i\le {\mathcal{G}}^{\ast }{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}},$$

where ${\nu }_i=\nu \left(\left\{x_i\right\}\right)$ and ${\mu }_i=\mu \left(\left\{x_i\right\}\right)$.

Now, for the regular part, there is a subsequence which satisfies

$$|u_{ϵ}{|}^{q(.)}dx\stackrel{\ast }{\rightharpoonup }{\nu }^{\ast }={|u|}^{q\left(x\right)}dx+\sum _{i\in I}{\nu }_i^{\ast }{\delta }_{x_i} \mathrm{or} {|u_{ϵ}-u|}^{q\left(x\right)}dx\stackrel{\ast }{\rightharpoonup }\sum _{i\in i}{\nu }_i^{\ast }{\delta }_{x_i}.$$

For $A\subseteq \Omega$ and $s>0$

$$\begin{array}{cc}\hfill {\int }_{A}hdx\le & \nu \left(A\right)\le \underset{ϵ\to 0}{lim\; inf}{\int }_A\frac{G\left(v_{ϵ}\right)}{{ϵ}^{q\left(x\right)}}dx,\hfill \\ \hfill \le & \underset{ϵ\to 0}{lim\; sup}{\int }_{A\cap \{|u_{ϵ}|<s\}}\frac{G\left(ϵu_{ϵ}\right)}{{ϵ}^{q\left(x\right)}{|u_{ϵ}|}^{q\left(x\right)}}|u_{ϵ}{|}^{q\left(x\right)}+c\underset{ϵ\to 0}{lim\; sup}{\int }_{A\cap \{|u_{ϵ}|\ge s\}}{|u_{ϵ}|}^{q\left(x\right)}dx.\hfill \end{array}$$

By Egoroff’s theorem, without loss of generality, we assume that ${lim}_{s\to 0}{s}^{-q\left(x\right)}G\left(s\right)$ converges uniformly to T on $\Omega$. Thus,

$$\begin{array}{cc}\hfill {\int }_{A}hdx\le & \underset{ϵ\to 0}{lim\; sup}{\int }_{A\cap \{|u_{ϵ}|<s\}}(T+o\left(1\right)){|u_{ϵ}|}^{q\left(x\right)}dx+\hfill \\ \hfill & c\underset{ϵ\to 0}{lim\; sup}\left({\int }_{A\cap \{|u_{ϵ}|\ge s\}}{|u|}^{q\left(x\right)}dx+{\int }_A{|u_{ϵ}-u|}^{q\left(x\right)}dx\right).\hfill \end{array}$$

On the other hand,

$$\underset{s\to \infty }{lim}|\{x\in A:|u_{ϵ}|\ge s\}|=0,$$

therefore,

$${\int }_{A}hdx\le {\int }_{A}Td{\nu }^{\ast }+c\sum _{x_i\in \overline{A}}{\nu }_i^{\ast }\le {\int }_{A}T{|u|}^{q\left(x\right)}dx+(G_0^{+}+c)\sum _{x_i\in \overline{A}}{\nu }_i^{\ast }.$$

By the Radon–Nikodym theorem, we deduce that $h\le {T|u|}^{q(.)}$ a.e. in $\Omega .$ □

The above theorem greatly helps us to pinpoint $\Gamma$-convergence limit $\overline{\mathcal{H}}$ and prove the main result.

Proof of lim sup inequality Theorem 1.

Let $(u_{ϵ},|\nabla u_{ϵ}{|}^{p(.)}dx)\stackrel{\tau }{\to }(u,\mu )$ in X, i.e., $u_{ϵ}\rightharpoonup u$ and $|\nabla u_{ϵ}{|}^{p(.)}dx\stackrel{\ast }{\rightharpoonup }\mu$ in measure. Let

$$H=\underset{ϵ\to 0}{lim\; sup}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ}).$$

Then, there exists a subsequence still called $u_{ϵ}$ such that ${ϵ}^{-q(.)}G\left(ϵu_{ϵ}\right)\stackrel{\ast }{\rightharpoonup }\nu$ in measure and

$$\underset{ϵ\to 0}{lim}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})=H.$$

By Theorem 3,

$$H={\int }_{\Omega }d\nu ={\int }_{\Omega }hdx+\sum _i{\nu }_i.$$

In addition, (19) and (22) give us

$$H\le {\int }_{\Omega }T{|u|}^{q(.)}dx+{\mathcal{G}}^{\ast }\sum _i{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}}=\overline{\mathcal{H}}(u,\mu ).$$

□

Proof of lim inf Inequality

In the proof of the lim inf inequality of Theorem 1(2), we discuss two cases: one in which $\mu$ has no atoms (Proposition 4) and second in which $\mu$ has a finite number of atoms (Propositions 5 and 6).

Proposition 4.

Let $(u,\mu =|\nabla u{|}^{p(.)}dx+\overline{\mu })\in X(\Omega )$, then

$$\underset{ϵ\to 0}{lim}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})=\overline{\mathcal{H}},$$

whenever, $(u_{ϵ},|\nabla u_{ϵ}{|}^{p(.)}dx)\stackrel{\tau }{\to }(u,\mu )$ in X.

Proof. 

Fix $x\in \Omega$, let us consider a sequence ${\left\{(u_{ϵ},{\mu }_{ϵ})\right\}}_{ϵ>0}$ from $X(\Omega )$, with ${\mu }_{ϵ}={|\nabla u_{ϵ}|}^{p(.)}dx$ and $(u_{ϵ},|\nabla u_{ϵ}{|}^{p(.)}dx)\stackrel{\mathcal{T}}{\to }(u,\mu )$. Then there is a subsequence $|u_{ϵ}{|}^{q(.)}\stackrel{\ast }{\rightharpoonup }{\nu }^{\ast }$ in $\mathcal{M}\left(\overline{\Omega }\right)$; furthermore, $\mu$ and ${\nu }^{\ast }$ have the same atoms and thus, $\nu$ is also atomless and ${\nu }^{\ast }={|u|}^{q(.)}$. In particular, ${\int }_{\Omega }|u_{ϵ}{|}^{q\left(x\right)}dx\to {\int }_{\Omega }{|u|}^{q\left(x\right)}dx$. In addition, $u_{ϵ}$ is weakly convergent to u in $L^{q(.)}(\Omega )$. Combining the two facts, we deduce that $u_{ϵ}\to u$ strongly in $L^{q(.)}(\Omega )$. Hence, there exists a subsequence ${\left\{(u_{ϵ},{\mu }_{ϵ})\right\}}_{ϵ>0}$ such that $u_{ϵ}\to u$ a.e. on $\Omega$ and ${lim}_{ϵ\to 0}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})={lim\; inf}_{ϵ\to 0}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})$. For a particular element $x\in \Omega$, if $u\left(x\right)\ne 0$ then $u_{ϵ}\left(x\right)\ne 0$, as $ϵ\to 0$, so

$${ϵ}^{-q\left(x\right)}G\left(ϵu_{ϵ}\left(x\right)\right)=(ϵ|u_{ϵ}{\left(x\right)|)}^{-q\left(x\right)}G\left(ϵu_{ϵ}\left(x\right)\right){|u_{ϵ}\left(x\right)|}^{q\left(x\right)},$$

and also we have

$$\underset{ϵ\to 0}{lim}(ϵ|u_{ϵ}{\left(x\right)|)}^{-q\left(x\right)}G\left(ϵu_{ϵ}\left(x\right)\right)|u_{ϵ}{\left(x\right)|}^{q\left(x\right)}=T\left(x\right){|u\left(x\right)|}^{q\left(x\right)}.$$

Now we discuss the case when $u\left(x\right)=0$. By using the growth condition on $G\left(s\right)$, we have the inequality

$${ϵ}^{-q\left(x\right)}G\left(ϵu_{ϵ}\right)\le c{|u_{ϵ}|}^{q\left(x\right)}.$$

which implies

$${ϵ}^{-q\left(x\right)}G\left(ϵu_{ϵ}\right)\to 0=T\left(x\right){|u\left(x\right)|}^{q\left(x\right)}.$$

Finally, using the Lebesgue dominated convergence theorem, we obtain

$$\underset{ϵ\to 0}{lim}{\int }_{\Omega }{ϵ}^{-q\left(x\right)}G\left(ϵu_{ϵ}\left(x\right)\right)={\int }_{\Omega }T\left(x\right){|u\left(x\right)|}^{q\left(x\right)}dx,$$

$$\underset{ϵ\to 0}{lim}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})={\int }_{\Omega }T\left(x\right){|u\left(x\right)|}^{q\left(x\right)}dx.$$

□

In order to discuss the atomic part, we first prove the lemma below.

Lemma 3.

For every pair $(u,\mu )\in X(\Omega )$, assume $q^{-}/p^{+}>1$, then we have the inequality $\overline{\mathcal{H}}(u,\mu )\le {\mathcal{G}}^{\ast }$. Additionally, $\overline{\mathcal{H}}(u,\mu )={\mathcal{G}}^{\ast }\iff (u,\mu )=(0,{\delta }_{x_0}),$ for some $x_0\in \overline{\Omega }.$

Proof. 

$$\begin{array}{cc}\hfill \overline{\mathcal{H}}(u,\mu )& ={\int }_{\Omega }T\left(x\right){|u|}^{q\left(x\right)}dx+{\mathcal{G}}^{\ast }\sum _i{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}},\hfill \\ \hfill & \le G_0^{+}{\int }_{\Omega }{|u|}^{q\left(x\right)}dx+{\mathcal{G}}^{\ast }\sum _i{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}},\hfill \\ \hfill & \le G_0^{+}{Smax\{\parallel \nabla u\parallel }_{L^{p\left(x\right)}(\Omega )}^{q^{+}}{,\parallel \nabla u\parallel }_{L^{p\left(x\right)}(\Omega )}^{q^{-}}\}+{\mathcal{G}}^{\ast }\sum _i{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}},\hfill \end{array}$$

since

$$1\ge \mu \left(\overline{\Omega }\right)\ge {\int }_{\Omega }{|\nabla u|}^{p(.)}dx+\sum _i{\mu }_i,$$

thus

$${max\{\parallel \nabla u\parallel }_{L^{p\left(x\right)}(\Omega )}^{q^{+}}{,\parallel \nabla u\parallel }_{L^{p\left(x\right)}(\Omega )}^{q^{-}}{\}=\parallel \nabla u\parallel }_{L^{p\left(x\right)}(\Omega )}^{q^{-}}\le {\left({\int }_{\Omega }{|\nabla u|}^{p(.)}dx\right)}^{q^{-}/p^{+}}.$$

Hence,

$$\overline{\mathcal{H}}(u,\mu )\le {\mathcal{G}}^{\ast }{\int }_{\Omega }{|\nabla u|}^{p(.)}dx+{\mathcal{G}}^{\ast }\sum _i{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}},$$

$$\overline{\mathcal{H}}(u,\mu )\le {\mathcal{G}}^{\ast }{\int }_{\Omega }{|\nabla u|}^{p(.)}dx+{\mathcal{G}}^{\ast }\sum _i{\mu }_i\le {\mathcal{G}}^{\ast },$$

and for the case when $u=0$, and $\mu ={\delta }_{x_0}$ for $x_0\in \overline{\Omega }$, we have the equality $\overline{\mathcal{H}}(u,\mu )={\mathcal{G}}^{\ast }$. □

Proposition 5.

For every open set ${\Omega }_0\subset \Omega$, for every $x\in \overline{{\Omega }_0}$ and for every $(u,\mu )\in X$ with $(u,\mu )=(0,{\delta }_x)$, there exists the Γ-limit of the sequence $\left\{{\mathcal{H}}_{ϵ}\right\}$ such that its restriction to ${\Omega }_0$ satisfies

$$(\Gamma -\underset{ϵ\to 0^{+}}{lim}{\mathcal{H}}_{ϵ})(0,{\delta }_x)={\mathcal{G}}^{\ast } \mathrm{in} X\left({\Omega }_0\right).$$

Proof. 

Let us fix the open set ${\Omega }_0\subset \Omega$ and also take ${ϵ}_s\to 0$ as $s\to \infty$. Using the compactness property of $\Gamma$-convergence, there exists a subsequence which we denote by $\left\{{ϵ}_s\right\}$ with a functional $\mathcal{H}:X\left({\Omega }_0\right)\to {\mathbb{R}}^{+}$, and

$${\mathcal{H}}_{{ϵ}_s}\left({\Omega }_0\right)\stackrel{\Gamma }{\to }\mathcal{H}\left({\Omega }_0\right).$$

Additionally, as $s\to \infty$, Lemma 1 gives

$$\underset{X\left({\Omega }_0\right)}{sup}{\mathcal{H}}_{{ϵ}_s}={\mathcal{G}}_{{ϵ}_s}\left({\Omega }_0\right)\to {\mathcal{G}}^{\ast }=\underset{X\left({\Omega }_0\right)}{max}\mathcal{H},$$

By Lemma 3 for $(u,\mu )\ne (0,{\delta }_{x_0})$ in ${\Omega }_0$ we obtain

$$\mathcal{H}(u,\mu )\le \underset{{ϵ}_s\to 0}{lim\; sup}\mathcal{H}(u_{{ϵ}_s},{\mu }_{{ϵ}_s})\le \overline{\mathcal{H}}(u,\mu )<{\mathcal{G}}^{\ast },$$

Hence, there exists some $x_0\in \overline{{\Omega }_0}$ such that $\mathcal{H}(0,{\delta }_{x_0})={\mathcal{G}}^{\ast }$.

Take a family of spheres $B_r^c\subset {\Omega }_0$, with center $c\in Q^n\cap {\Omega }_0$ and radius $r\in Q$. We can find by diagonalization a subsequence denoted by $\left\{{ϵ}_s\right\}$, a functional $\mathcal{H}$ such that ${\mathcal{H}}_{{ϵ}_s}\left(B_r^c\right)\stackrel{\Gamma }{\to }\mathcal{H}\left(\overline{B_r^c}\right)$ for each $r\in Q$ and for each $c\in Q^n\cap {\Omega }_0,$ and ${\mathcal{H}}_{{ϵ}_s}\left({\Omega }_0\right)\stackrel{\Gamma }{\to }\mathcal{H}\left(\overline{{\Omega }_0}\right).$ Following the similar method as before, for each sphere $B_r^c$, we can find $x_r^c\in B_r^c$ such that

$$\mathcal{H}(0,{\delta }_{x_r^c})={\mathcal{G}}^{\ast }.$$

Now as $r\to 0$, then $x_r^c\to c$ and the fact that $\Gamma$-limit is upper semicontinuous on $\overline{{\Omega }_0}$, we obtain

$$\underset{r\to 0}{lim\; sup}\mathcal{H}(0,{\delta }_{x_r^c})\le \mathcal{H}(0,{\delta }_q),$$

and by combining, we obtain

$${\mathcal{G}}^{\ast }=\underset{r\to 0}{lim}H(0,{\delta }_{x_r^c})\le \underset{r\to 0}{lim\; sup}\mathcal{H}(0,{\delta }_{x_r^c})\le \mathcal{H}(0,{\delta }_c),$$

which implies

$$\mathcal{H}(0,{\delta }_c)={\mathcal{G}}^{\ast }.$$

Again, using the fact that $\Gamma$-limit is upper semicontinuous and the density of $Q^n\cap {\Omega }_0$ in $\overline{{\Omega }_0}$, for each $x\in \overline{{\Omega }_0}$ with $(u,\mu )=(0,{\delta }_x),$ we have

$$\mathcal{H}(0,{\delta }_x)={\mathcal{G}}^{\ast }.$$

The above process is independent of the selection of sequence; hence, for each pair $(u,\mu )=(0,{\delta }_x)$ where $x\in \overline{{\Omega }_0}$, there always exists a $\Gamma$-limit $\mathcal{H}$ and $\mathcal{H}(0,{\delta }_x)={\mathcal{G}}^{\ast }$. □

Proposition 6.

Let $(u,\mu )=(0,{\sum }_{i=0}^N{\mu }_i{\delta }_{x_i})$ be a pair in $X(\Omega )$. Then

$${\mathcal{H}}_{ϵ}\stackrel{\Gamma }{\to }\overline{\mathcal{H}}.$$

Proof. 

We will prove our result for $N=1$, $\mu ={\sum }_{i=0}^1{\mu }_i{\delta }_{x_i},$ where ${\mu }_i\in (0,1)$ with ${\sum }_{i=0}^1{\mu }_i\le 1$. We introduce first ${\Omega }_i=\Omega \bigcap B_{r_i}\left(x_i\right)$, where $dist({\Omega }_0,{\Omega }_1)>0$ for $i\in \{0,1\}$. By Proposition 5, there are sequences $(u_{ϵ}^i,{\mu }_{ϵ}^i)$ in $X\left({\Omega }_i\right)$ such that $(u_{ϵ}^i,{\mu }_{ϵ}^i)\stackrel{\mathcal{T}}{\to }(0,{\delta }_{x_i})$, where ${\mu }_{ϵ}^i={|\nabla u_{ϵ}^i|}^{p\left(x\right)}dx$ and $\overline{\mathcal{H}}(0,{\delta }_{x_i},\overline{{\Omega }_i})={\mathcal{G}}^{\ast }$.

$$\underset{ϵ\to 0^{+}}{lim\; inf}{\mathcal{H}}_{ϵ}(u_{ϵ}^i,{\mu }_{ϵ}^i,{\Omega }_i)\ge \overline{\mathcal{H}}(0,{\delta }_{x_i},\overline{{\Omega }_i})={\mathcal{G}}^{\ast },$$

which gives

$$\underset{ϵ\to 0^{+}}{lim}{\int }_{{\Omega }_i}{ϵ}^{-q\left(x\right)}G\left(ϵu_{ϵ}^i\right)dx={\mathcal{G}}^{\ast },i\in \{0,1\}.$$

Let us set $u_{ϵ}={\left({\mu }_0\right)}^{1/p\left(x_0\right)}{u}_{ϵ}^0+{\left({\mu }_1\right)}^{1/p\left(x_1\right)}{u}_{ϵ}^1$ with ${\mu }_{ϵ}={|\nabla u_{ϵ}|}^{p(.)}$. Due to the fact that $x_0\ne x_1$ and that the supports of $u_{ϵ}^0$ and $u_{ϵ}^1$ are disjoint for small $ϵ$, it is to be noted that the functions $u_{ϵ}$ for every given $\varphi \in C\left(\overline{\Omega }\right)$ satisfy that

$${\int }_{\Omega }{\varphi |\nabla u|}^{p(.)}dx={\int }_{{\Omega }_0}{\mu }_0^{p\left(x\right)/p\left(x_0\right)}\varphi |\nabla u_{ϵ}^0{|}^{p(.)}dx+{\int }_{{\Omega }_1}{\mu }_1^{p\left(x\right)/p\left(x_1\right)}\varphi {|\nabla u_{ϵ}^1|}^{p(.)}dx,$$

$$\underset{ϵ\to 0}{lim}{\int }_{\Omega }\varphi {|\nabla u|}^{p(.)}dx={\mu }_0\varphi \left(x_0\right)+{\mu }_1\varphi \left(x_1\right)={\int }_{\Omega }\varphi d\mu ,$$

which implies specifically that

$$\underset{ϵ\to 0}{lim}{\int }_{\Omega }{|\nabla u|}^{p(.)}dx={\mu }_0+{\mu }_1=\mu \left(\overline{\Omega }\right)<1.$$

Thus, we obtain that the pair $(u_{ϵ},|\nabla u_{ϵ}{|}^{p(.)}dx)$ is the part of the set $X(\Omega )$ for $ϵ$ sufficiently small and further $(u_{ϵ},|\nabla u_{ϵ}{|}^{p(.)}dx)\stackrel{\mathcal{T}}{\to }(0,\mu )$ when $ϵ\to 0$. Finally, again using the fact that supports of $u_{ϵ}^0$ and $u_{ϵ}^1$ are disjoint, we obtain

$$\begin{array}{cc}\hfill {\mathcal{H}}_{ϵ}(u_{ϵ},|\nabla u_{ϵ}{|}^{p(.)})=& {\int }_{\Omega }\frac{G\left(ϵu_{ϵ}\right)}{{ϵ}^{q(.)}}dx,\hfill \\ \hfill =& {\int }_{{\Omega }_0}{\mu }_0^{q(.)/p\left(x_0\right)}\frac{G\left(ϵ{\mu }_0^{1/p\left(x_0\right)}{u}_{ϵ}^0\right)}{{{\mu }_0^{q(.)/p\left(x_0\right)}ϵ}^{q(.)}}dx+{\int }_{{\Omega }_1}{\mu }_1^{q(.)/p\left(x_1\right)}\frac{G\left(ϵ{\mu }_1^{1/p\left(x_1\right)}{u}_{ϵ}^1\right)}{{{\mu }_1^{q(.)/p\left(x_1\right)}ϵ}^{q(.)}}dx,\hfill \\ \hfill & \to {\mu }_0^{\frac{q\left(x_0\right)}{p\left(x_0\right)}}{\mathcal{G}}^{\ast }+{\mu }_1^{\frac{q\left(x_1\right)}{p\left(x_1\right)}}{\mathcal{G}}^{\ast }=\overline{\mathcal{H}}(u,\mu ).\hfill \end{array}$$

□

The next lemma shows that the proof of the lim inf inequality of Theorem 1(2) for measures of disjoint support is sufficient for the general case.

Lemma 4.

If the lim inf inequality of Theorem 1(2) holds for $(u,\mu )\in X(\Omega )$ which satisfies

• $\mu \left(\overline{\Omega }\right)<1,$
• $\mu ={|\nabla u|}^{p(.)}dx+\overline{\mu }+{\sum }_{i=0}^n{\mu }_i{\delta }_{x_i},$
• $dist(\overline{supp(|u|+\overline{\mu })},{\cup }_{i=0}^n\left\{x_i\right\})>0,$

then the lim inf inequality of Theorem 1(2) holds for every $(u,\mu )\in X(\Omega )$.

Proof. 

Take $(u,\mu )\in X(\Omega )$ such that $\mu ={|\nabla u|}^{p(.)}dx+\overline{\mu }+{\sum }_{i=0}^n{\mu }_i{\delta }_{x_i}$ and $\mu \left(\overline{\Omega }\right)<1$. Our target is to make a sequence which satisfies the following conditions:

• $(u_{\sigma },{\mu }_{\sigma })\in X(\Omega )$ and ${\mu }_{\sigma }={|\nabla u_{\sigma }|}^{p(.)}dx+{\overline{\mu }}_{\sigma }+{\sum }_{i=0}^n{\mu }_i{\delta }_{x_i},$
• ${\mu }_{\sigma }\left(\overline{\Omega }\right)<1$ as $\sigma$ goes to 0,
• $dist(\overline{supp(|u_{\sigma }|+{\overline{\mu }}_{\sigma })},{\cup }_{i=0}^n\left\{x_i\right\})>0,$
• $u_{\sigma }\to u$ in $L^{q(.)}(\Omega )$ and $(u_{\sigma },{\mu }_{\sigma })\stackrel{\mathcal{T}}{\to }(u,\mu )$,
• ${lim}_{\sigma \to 0}\mathcal{H}(u_{\sigma },{\mu }_{\sigma })=\overline{\mathcal{H}}(u,\mu ).$

Define $B_{\sigma }\left(x_i\right)\cap \Omega =B_{\sigma }^i$ for $\sigma >0$ and $i\in \{0,\dots ,N\}$ and choose cut off functions ${\varphi }_{\sigma }$, with $0\le {\varphi }_{\sigma }\le 1$, ${\varphi }_{\sigma }=1$ in $\Omega \backslash {\cup }_{i=0}^n{B}_{2\sigma }^i,{\varphi }_{\sigma }=0$ in ${\cup }_{i=0}^n{B}_{\sigma }^i$, $|\nabla {\varphi }_{\sigma }|\le 1/\sigma$. And further set

$$\begin{gathered} (u_{\sigma },{\mu }_{\sigma })=\left(u{\varphi }_{\sigma },{|\nabla u{\varphi }_{\sigma }|}^{p(.)}dx+\overline{\mu }{\varphi }_{\sigma }+\sum _{i=0}^n{\mu }_i{\delta }_{x_i}\right), \\ =\left({u,|\nabla u|}^{p(.)}dx+\overline{\mu }+{\sum }_{i=0}^n{\mu }_i{\delta }_{x_i}\right), \end{gathered}$$

By similar steps as in [6], we have the following deductions: $u_{\sigma }\in W_0^{1,p(.)}(\Omega )$ and ${\mu }_{\sigma }\in M\left(\overline{\Omega }\right)$, $dist(\overline{supp(|u_{\sigma }|+{\overline{\mu }}_{\sigma })},{\cup }_{i=0}^n\left\{x_i\right\})\ge \sigma >0.$

$${\int }_{\Omega }|u{\varphi }_{\sigma }{-u|}^{q(.)}dx\le \sum _{i=0}^n{\int }_{B_{2\sigma }}{|u|}^{q(.)}dx\to 0,$$

as $\sigma \to 0$. Hence $u_{\sigma }\to u$ in $L^{q(.)}(\Omega )$. Further for every $\psi \in {\mathcal{C}}^0\left(\overline{\Omega }\right),$ we have ${\int }_{\overline{\Omega }}\psi d{\mu }_{\sigma }\to {\int }_{\overline{\Omega }}\psi d\mu$ which implies ${\mu }_{\sigma }\rightharpoonup \mu$ as measures in the space $M\left(\overline{\Omega }\right)$ when $\sigma$ goes to 0. Combining them, we have $(u_{\sigma },{\mu }_{\sigma })\stackrel{\mathcal{T}}{\to }(u,\mu )$, and as $\mu \left(\overline{\Omega }\right)<1$, which implies that ${\mu }_{\sigma }$ satisfies condition (2). In particular $(u_{\sigma },{\mu }_{\sigma })\in X(\Omega )$. The last condition is satisfied by the definition of the $\Gamma$-limit and $u_{\sigma }\to u$ in $L^{q(.)}(\Omega )$. Thus, we have

$$\underset{\sigma \to 0}{lim\; inf}{\mathcal{H}}_{\sigma }(u_{\sigma },{\mu }_{\sigma })\ge \overline{\mathcal{H}}(u_{\sigma },{\mu }_{\sigma }),$$

when $\sigma >0$. By the upper semicontinuity of the $\Gamma$-limit, we obtain

$$\underset{\sigma \to 0}{lim\; inf}{\mathcal{H}}_{\sigma }(u_{\sigma },{\mu }_{\sigma })\ge \underset{\sigma \to 0}{lim\; sup}\underset{\sigma \to 0}{lim\; inf}{\mathcal{H}}_{\sigma }(u_{\sigma },{\mu }_{\sigma })\ge \underset{\sigma \to 0}{lim}\overline{\mathcal{H}}(u_{\sigma },{\mu }_{\sigma })=\overline{\mathcal{H}}(u,\mu ),$$

where $(u,\mu )\in X(\Omega )$ and $\mu \left(\overline{\Omega }\right)<1$, $\mu ={|\nabla u|}^{p(.)}dx+\overline{\mu }+{\sum }_{i=0}^n{\mu }_i{\delta }_{x_i}$.

Now consider a pair $(u,\mu )\in X(\Omega )$ such that $\mu \left(\overline{\Omega }\right)<1$. Define a sequence $(u_m,{\mu }_m)$ such that

$$u_m=u,{\mu }_m={|\nabla u|}^{p(.)}+\overline{\mu }+\sum _{i=0}^m{\mu }_i{\delta }_{x_i}.$$

$m\ge 0$.

Obviously, $(u_m,{\mu }_m)\in X(\Omega )$ and $(u_m,{\mu }_m)\stackrel{\mathcal{T}}{\to }(u,\mu ),$ as $m\to +\infty$. Using the above steps for the pair $(u_m,{\mu }_m)$, we obtain

$$\underset{m\to +\infty }{lim\; inf}{\mathcal{H}}_m(u_m,{\mu }_m)\ge \overline{\mathcal{H}}(u_m,{\mu }_m),$$

for $m>0$. Again, applying the upper semicontinuity of the $\Gamma$-limit, we have

$$\underset{m\to +\infty }{lim\; inf}{\mathcal{H}}_m(u_m,{\mu }_m)\ge \underset{m\to +\infty }{lim\; sup}\underset{m\to +\infty }{lim\; inf}{\mathcal{H}}_m(u_m,{\mu }_m)\ge \underset{m\to +\infty }{lim}\mathcal{H}(u_m,{\mu }_m)=\overline{\mathcal{H}}(u,\mu ).$$

Lastly, we study for general pairs $(u,\mu )\in X(\Omega )$ for $\rho >0$, defined by

$$u_{\rho }=\frac{u}{1+\rho }, {\mu }_{\rho }=\frac{\mu }{{(1+\rho )}^{p(.)}}.$$

which then becomes

$${\mu }_{\rho }={|\nabla u_{\rho }|}^{p(.)}+\frac{\overline{\mu }}{{(1+\rho )}^{p(.)}}+\sum _{i=0}^{+\infty }\frac{{\mu }_i}{{(1+\rho )}^{p(.)}}{\delta }_{x_i}.$$

It is clear that $(u_{\rho },{\mu }_{\rho })\in X(\Omega )$ and ${\mu }_{\rho }(\Omega )<1$, $(u_{\rho },{\mu }_{\rho })\stackrel{\mathcal{T}}{\to }(u,\mu )$, as $\rho \to 0.$ On the lines of the previous steps, we obtain

$$\underset{\rho \to 0}{lim\; inf}{\mathcal{H}}_{\rho }(u_{\rho },{\mu }_{\rho })\ge \overline{\mathcal{H}}(u_{\rho },{\mu }_{\rho })={\int }_{\Omega }\frac{T\left(x\right)}{{(1+\rho )}^{q(.)}}{|u|}^{q(.)}+\frac{{\mathcal{G}}^{\ast }}{{(1+\rho )}^{q(.)}}\sum _i{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}}.$$

Again applying the upper semicontinuity of the $\Gamma$-limit, we have

$$\underset{\rho \to 0}{lim\; inf}{\mathcal{H}}_{\rho }(u_{\rho },{\mu }_{\rho })\ge \underset{\rho \to 0}{lim\; sup}\underset{\rho \to 0}{lim\; inf}{\mathcal{H}}_{\rho }(u_{\rho },{\mu }_{\rho })\ge \underset{\rho \to 0}{lim}\mathcal{H}(u_{\rho },{\mu }_{\rho })=\overline{\mathcal{H}}(u,\mu ).$$

□

Now we are in a position to prove the main result.

Proof of Theorem 1.

To complete the proof, it remains to show the lim inf inequality ${lim\; inf}_{ϵ\to 0}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})\ge \overline{\mathcal{H}}$ for the pair $(u,\mu )\in X(\Omega )$, satisfying the conditions of Lemma 4, given as follows:

• $\mu \left(\overline{\Omega }\right)<1,$
• $\mu ={|\nabla u|}^{p(.)}dx+\overline{\mu }+{\sum }_{i=0}^n{\mu }_i{\delta }_{x_i},$
• $dist(\overline{supp(|u|+\overline{u})},{\cup }_{i=0}^n\left\{x_i\right\})>0.$

Let us denote ${\mu }^A={|\nabla u|}^{p(.)}dx+\overline{\mu }$ and ${\mu }^B={\sum }_{i=0}^n{\mu }_i{\delta }_{x_i}=\mu -{\mu }^A$. In addition, consider two open sets A and B in $\Omega$, with $\overline{supp(|u|+\overline{\mu })}\subset \overline{A}$, $\overline{supp\left({\mu }^B\right)}\subset \overline{B}$ and $\overline{A}\cap \overline{B}=\varnothing$.

In view of Proposition 4, there exists a sequence of pairs $(u_{ϵ}^A,{\mu }_{ϵ}^A)\in X(\Omega )$, where $u_{ϵ}^A\in W_0^{1,p(.)}\left(A\right)$, ${\mu }_{ϵ}^A={|\nabla u_{ϵ}^A|}^{p(.)}dx$ with ${\mu }_{ϵ}^A\left(\overline{\Omega }\right)<1$ for small $ϵ$, and

$$(u_{ϵ}^A,{\mu }_{ϵ}^A)\stackrel{\mathcal{T}}{\to }(u,{\mu }^A)\mathrm{when}ϵ\to 0,$$

and also,

$${\mathcal{H}}_{ϵ}(u_{ϵ}^A,{\mu }_{ϵ}^A)\to \overline{\mathcal{H}}(u,{\mu }^A)\mathrm{when}ϵ\to 0.$$

(24)

In the same way, using Proposition 6, it follows that there exists a sequence of pairs $(u_{ϵ}^B,{\mu }_{ϵ}^B)\in X(\Omega )$, where $u_{ϵ}^B\in W_0^{1,p(.)}\left(B\right)$, ${\mu }_{ϵ}^B={|\nabla u_{ϵ}^B|}^{p(.)}dx$ and ${\mu }_{ϵ}^B\left(\overline{\Omega }\right)<1$ for $ϵ$ small enough, and

$$(u_{ϵ}^B,{\mu }_{ϵ}^B)\stackrel{\mathcal{T}}{\to }(0,{\mu }^B)\mathrm{when}ϵ\to 0,$$

and also,

$${\mathcal{H}}_{ϵ}(u_{ϵ}^B,{\mu }_{ϵ}^B)\to \overline{\mathcal{H}}(0,{\mu }^B)\mathrm{when}ϵ\to 0.$$

(25)

Set now $u_{ϵ}=u_{ϵ}^A+u_{ϵ}^B$ with ${\mu }_{ϵ}={\mu }_{ϵ}^A+{\mu }_{ϵ}^B$. Since $supp\left(u_{ϵ}^A\right)\cap supp\left(u_{ϵ}^B\right)=\varnothing$, $(u_{ϵ},{\mu }_{ϵ})\in X(\Omega )$ with $u_{ϵ}\in W_0^{1,p(.)}$, ${\mu }_{ϵ}=|\nabla u_{ϵ}{|}^{p(.)}=|\nabla u_{ϵ}^A{|}^{p(.)}+{|\nabla u_{ϵ}^B|}^{p(.)}$ and for small $ϵ$ we have

$${\mu }_{ϵ}\left(\overline{\Omega }\right)={\int }_{\Omega }|\nabla u_{ϵ}^A{|}^{p(.)}dx+{\int }_{\Omega }{|\nabla u_{ϵ}^B|}^{p(.)}dx={\mu }^A\left(\overline{\Omega }\right)+{\mu }^B\left(\overline{\Omega }\right)=\mu \left(\overline{\Omega }\right)<1.$$

By (24) and (25), we have finally

$${\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})={\int }_{\Omega }{ϵ}^{-q\left(x\right)}G\left(ϵu_{ϵ}\right)dx={\int }_A{ϵ}^{-q\left(x\right)}G\left(ϵu_{ϵ}^A\right)dx+{\int }_B{ϵ}^{-q\left(x\right)}G\left(ϵu_{ϵ}^B\right)dx,$$

and taking the limit as $ϵ\to 0,$

$$\underset{ϵ\to 0}{lim}{\mathcal{H}}_{ϵ}(u_{ϵ},{\mu }_{ϵ})={\int }_{\Omega }T\left(x\right){|u|}^{q\left(x\right)}dx+{\mathcal{G}}^{\ast }\sum _{i=0}^n{\mu }_i^{\frac{q\left(x_i\right)}{p\left(x_i\right)}}=\overline{\mathcal{H}}(u,\mu ).$$

□

Now we give the proof of the second main result, which follows directly from the $\Gamma$-convergence result and Lemma 3.

Proof Theorem 2.

We saw in the $\Gamma$-convergence result and the Lemma 3 that, if $\left\{(u_{ϵ},{\mu }_{ϵ})\right\}$ is the maximizing sequence for ${\mathcal{H}}_{ϵ}$, then $\left\{(u_{ϵ},{\mu }_{ϵ})\right\}$ converges to $(0,{\delta }_x)$, where $x\in \overline{\Omega }$ and every pair $(0,{\delta }_x)$ is a maximizer for the $\Gamma$-limit $\overline{\mathcal{H}}$. □

5. Conclusions

We establish the $\Gamma$-convergence of low energy extremals for a class of general functions in settings of variable exponent Lebesgue spaces. This work will help to determine low energy solutions. Normally, they form a spike near concentration points in the domain.

In the future, we can work on the geometric shape of these spikes with the help of Robin functions. Another line of research is to study these problems in fractional Sobolev spaces [27,28,29] and fractional-order systems [30,31].