Abstract
In this paper, we are concerned with the Neumann problem for a class of quasilinear stationary Kirchhoff-type potential systems, which involves general variable exponents elliptic operators with critical growth and real positive parameter. We show that the problem has at least one solution, which converges to zero in the norm of the space as the real positive parameter tends to infinity, via combining the truncation technique, variational method, and the concentration–compactness principle for variable exponent under suitable assumptions on the nonlinearities.
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1 Introduction
In this article, we are concerned with the existence and asymptotic behavior of nontrivial solutions for the following class of nonlocal quasilinear elliptic systems:
for \(i=1,2,\dots ,n\) (\(n\in \mathbb{N}^{\ast }\)), where \(\Omega \subset \mathbb{R}^{N}\) (\(N\geq 2\)) is a bounded domain with smooth boundary ∂Ω, \(\mathfrak{N}_{i}\) is the outward normal vector field on ∂Ω, λ is a positive parameter, \(\nabla F=(F_{u_{1}},\dots ,F_{u_{2}})\) is the gradient of a \(C^{1}\)-function \(F: \mathbb{R}^{N}\times \mathbb{R}^{n}\to \mathbb{R}\), \(p_{i}, q_{i}, w_{i},s_{i} \in C(\overline{\Omega})\), and \(\ell _{i}\in C(\partial \Omega )\) are such that
and
for all \(x\in \overline{\Omega}\), where functions \(w_{i}\) are given by condition \((\boldsymbol{F}_{4})\) below, \(p_{i}^{-}:= \inf_{x\in \overline{\Omega}}p_{i}(x)\), \(p_{i}^{+}:= \sup_{x\in \overline{\Omega}}p_{i}(x)\), and analogously to \(w_{i}^{-}\), \(w_{i}^{+}\), \(q_{i}^{-}\), \(q_{i}^{+}\), \(h_{i}^{-}\), \(h_{i}^{+}\), \(s_{i}^{-}\), \(s_{i}^{+}\), \(\ell _{i}^{-}\) and \(\ell _{i}^{+}\), with \(h_{i}(x) =(1-\mathcal{K}(k^{3}_{i})) p_{i}(x)+ \mathcal{K}(k^{3}_{i})q_{i}(x)\), where \(k^{3}_{i}\) is given by condition \((\boldsymbol{A}_{2})\) below, and
for all \(x\in \overline{\Omega}\), and the function \(\mathcal{K}:\mathbb{R}_{0}^{+}\to \lbrace 0,1 \rbrace \) is defined by
In addition, we consider both \(\mathcal{C}^{1}_{h_{i}}\) and \(\mathcal{C}^{2}_{h_{i}} \) as nonempty disjoint sets, which are respectively defined by
The operators \(\mathcal{A}_{j_{i}} : X_{i}\to \mathbb{R}^{n}\), for \(j=1\) or 2, and the operators \(\mathcal{B}_{i} : X_{i}\to \mathbb{R}\) are respectively defined by
for all \(1\leq i\leq n\), where \(X_{i}:= W^{1,h_{i}(x)}(\Omega )\cap W^{1,p_{i}(x)}(\Omega )\) is a Banach space, and functions \(A_{j_{i}}\) are defined by \(A_{j_{i}}(t)= \int _{0}^{t}a_{j_{i}}(k)\,d k\), where functions \(a_{j_{i}}\) are described in condition \((\boldsymbol{A}_{1})\).
In what follows, we shall consider the functions \(a_{j_{i}}\) satisfying the following assumptions for all \(i\in \{1,2,\dots ,n\}\) and \(j\in \{1,2\}\):
- \((\boldsymbol{A}_{1})\):
-
\(a_{j_{i}}:\mathbb{R}^{+} \to \mathbb{R}^{+}\) are of class \(C^{1}\).
- \((\boldsymbol{A}_{2})\):
-
There exist positive constants \(k_{j_{i}}^{0}\), \(k_{j_{i}}^{1}\), \(k_{j_{i}}^{2}\) and \(k_{i}^{3}\) for all \(i\in \{1,2,\dots ,n\}\), \(j\in \{1,2\}\) such that
$$ k_{j_{i}}^{0} + \mathcal{K}\bigl(k_{i}^{3} \bigr)k_{j_{i}}^{2} \xi ^{ \frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}} \leq a_{j_{i}}( \xi ) \leq k_{j_{i}}^{1} + k_{i}^{3}\xi ^{\frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}},\quad \text{for all } \xi \geq 0 \text{ and a.e. } x\in \overline{\Omega}. $$ - \((\boldsymbol{A}_{3})\):
-
There exists \(c_{i}>0\) for all \(1\leq i\leq n\) such that
$$ \begin{gathered} \min \biggl\lbrace a_{j_{i}}\bigl(\xi ^{p_{i}(x)}\bigr) \xi ^{p_{i}(x)-2}, a_{j_{i}}\bigl( \xi ^{p_{i}(x)}\bigr)\xi ^{p_{i}(x)-2} +\xi \frac{\partial (a_{j_{i}}(\xi ^{p_{i}(x)})\xi ^{p_{i}(x)-2})}{\partial \xi } \biggr\rbrace \geq c_{i}\xi ^{p_{i}(x)-2},\\ \quad \text{for a.e. } x\in \overline{\Omega} \text{ and all } \xi \geq 0. \end{gathered} $$ - \((\boldsymbol{A}_{4})\):
-
There exist positive constants \(\beta _{j_{i}}\) and \(\gamma _{i}\) for all \(i\in \lbrace 1,2,\dots ,n \rbrace \), \(j\in \{1,2\}\) such that
$$ A_{j_{i}}(\xi )\geq \frac{1}{\beta _{j_{i}}}a_{j_{i}}(\xi )\xi\quad \text{with } h_{i}^{+}< \gamma _{i} < s_{i}^{-} \text{ and } \frac{q_{i}^{+}}{p_{i}^{+}}\leq \beta _{j_{i}} < \frac{\gamma _{i}}{p_{i}^{+}}, \text{ for all } \xi \geq 0. $$ - \((\boldsymbol{M})\):
-
\(M_{i}: \mathbb{R}^{+} \rightarrow \mathbb{R}\) are continuous and increasing functions such that \(M_{i}(t)\geq M_{i}(0)=\mathfrak{M}_{i}^{0}>0 \), for all \(t \geq 0\), \(i\in \lbrace 1,2,\dots ,n \rbrace \).
As it is well known, there are many examples of functions \(M_{i} \) that satisfy assumption \((\boldsymbol{M})\), for example,
In particular, when \(\mathfrak{M}_{1}^{0}=0\) and \(\mathfrak{B}_{1}>0\), the Kirchhoff equation associated with \(M_{1}\) is said to be degenerate. On the other hand, when \(\mathfrak{M}_{1}^{0}>0\) and \(\mathfrak{B}_{1}\geq 0\), the Kirchhoff equation associated with \(M_{1}\) is said to be nondegenerate. In this case, when \(\mathfrak{B}_{1}=0\), the Kirchhoff equation associated with \(M_{1}\) (is a constant) reduces to a local quasilinear elliptic problem.
The study of differential equations and variational problems driven by nonhomogeneous differential operators has received extensive attention and has been extensively investigated, see, e.g., Papageorgiou et al. [42]. This is due to their ability to model many physical phenomena. It should be noted that the \(p(x)\)-Laplacian operator is a special case of the divergence form operator \(\operatorname{div} ( \mathcal{B}_{j_{i}}(\nabla u_{i}) )\). The natural functional framework for this operator is described by the Sobolev space with a variable exponent \(W^{1,p(x)}\).
In recent decades, there has been a particular focus on variable exponent Lebesgue and Sobolev spaces, \(L^{p(x)}\) and \(W^{1,p(x)}\), where p is a real function, e.g., Rǎdulescu and Repovš [45]. Traditional Lebesgue spaces \(L^{p}\) and Sobolev spaces \(W^{1,p}\) with constant exponents have proven insufficient to tackle the complexities of nonlinear problems in the applied sciences and engineering. To address these limitations, the use of variable exponent Lebesgue and Sobolev spaces has been on the rise in recent years.
This area of research reflects a new type of physical phenomena, such as electrorheological fluids, or “smart fluids,” which can exhibit dramatic changes in mechanical properties in response to an electromagnetic field. These and other nonhomogeneous materials require the use of variable exponent Lebesgue and Sobolev spaces, where the exponent p is allowed to vary. Moreover, variable exponent Lebesgue and Sobolev spaces have found a wide range of applications, from image restoration and processing to fields such as thermorheological fluids, mathematical biology, flow in porous media, polycrystal plasticity, heterogeneous sand pile growth, and fluid dynamics. For a comprehensive overview of these and other applications, see, e.g., Chen et al. [15], Diening et al. [20, 21], Halsey [28], Rǎdulescu [44], Ružic̆ka [46, 47], and the references therein.
Furthermore, every single equation of the system (1.1) is a generalization of the stationary problem of the first model introduced by Kirchhoff [33] in 1883 and having the following form:
where the parameters ρ, h, \(\rho _{0}\), t, L, E are all constants which respectively have some physical meaning, which is an extension of the classical D’Alembert wave equation, by considering the effect of the change in the length of a vibrating string.
Nearly a century later, in 1978, Jacques-Louis Lions [35] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with an external force term which was written as
Later on, many interesting results have been obtained by Caristi et al. [9], Dai and Hao [19], Ma [37]; see also the references therein. The main difficulty in studying these equations appears to be due to the fact that they do not satisfy a pointwise identity any longer. It is generated by having the term containing \(M_{i}\) in the equations, and it makes (1.1) a nonlocal problem. The nonlocal problem models arise in the description of biological systems and also various physical phenomena, where u describes a process that depends on the average of itself, such as population density. For more references on this subject, we refer the interested reader to Ambrosio et al. [3], Arosio and Pannizi [5], Cavalcanti et al. [10], Chipot and Lovat [17], He et al. [31], Corrêa and Nascimento [18], Yang and Zhou [50], Wang et al. [48], and the references therein. On the one hand, the differential equations with constant or variable critical exponents in bounded or unbounded domains have attracted increasing attention recently. They were first discussed in the seminal paper by Brezis and Nirenberg [8] in 1983, which treated Laplacian equations. Since then, there have been extensions of [8] in many directions.
One of the main features of elliptic equations involving critical growth is the lack of compactness arising in connection with the variational approach. In order to overcome the lack of compactness, Lions [36] established the method using the so-called concentration compactness principle (CCP, for short) to show that a minimizing sequence or a Palais–Smale ((PS), for short) sequence is precompact. Afterward, the variable exponent version of the Lions concentration–compactness principle for a bounded domain was independently obtained in Bonder et al. [6, 7] and Fu [26], and for an unbounded domain in Fu [27]. Since then, many authors have applied these results to study critical elliptic problems involving variable exponents, see, e.g., Alves et al. [1, 2], Chems Eddine et al. [12–14], Fang and Zhang [24], Hurtado et al. [32], Mingqi et al. [39], Zhang and Fu [51].
When \(M_{i}\) satisfies conditions \(a_{1_{i}}\equiv 1\) (with \(k^{1}_{i}=1\), and \(k^{2}_{i}>0\) and \(k^{3}_{i}=0\)), and \(a_{2_{i}}\equiv 0\), Chems Eddine [12] proved the existence of nontrivial weak solutions for the following class of Kirchhoff-type potential systems with Dirichlet boundary conditions:
for \(1\leq i\leq n\) (\(n\in \mathbb{N}^{\ast}\)), where \(p_{i},q_{i} : \overline{\Omega}\to \mathbb{R} \) are Lipschitz continuous functions such that \(1\leq q_{i} (x)\leq p_{i}^{\ast}(x)\) for all x in Ω and the potential function F satisfies mixed and subcritical growth conditions. Following that, Chems Eddine [11] established the existence of infinitely many solutions for the following system:
for \(i=1,2,\dots ,n\) (\(n\in \mathbb{N} \)), where \(\lbrace x \in \Omega , s_{i} (x) =p_{i}^{\ast}(x) \rbrace \) is nonempty and \(F\in C^{1}(\Omega \times \mathbb{R}^{n},\mathbb{R})\) satisfies some mixed and subcritical growth conditions. Next, Chems Eddine and Ragusa [14] dealt with cases when the above class of Kirchhoff-type potential systems is considered under Neumann boundary conditions with two critical exponents, and established the existence and multiplicity of solutions.
Our objective in this article is to show the existence of nontrivial solutions for the nonlocal problem (1.1). As we shall see in this paper, there are some substantial difficulties in our situation, which can be summed up in three main problems. First, the assumption \((\boldsymbol{M})\) provides only a positive lower bound for the Kirchhoff functions \(M_{i}\) near zero, creating serious mathematical technical difficulties, to overcome which we need to do a truncation of the Kirchhoff functions \(M_{i}\) to obtain a priori estimates of the boundedness from above, and thus obtain a new auxiliary problem, thereafter, we do another truncation to control the energy functional corresponding to the auxiliary problem. The second difficulty in solving problem (1.1) is the lack of compactness which can be illustrated by the fact that the embeddings \(W^{1,h(x)}(\Omega ) \hookrightarrow L^{h^{*}(x)}(\Omega )\) and \(W^{1,h(x)}(\Omega ) \hookrightarrow L^{h^{\partial}(x)}(\partial \Omega )\) are no longer compact and, to overcome this difficulty, we use two versions of Lions’s principle for the variable exponent, extended by Bonder et al. [6, 7]. Then by combining the variational method and Mountain Pass Theorem, we obtain the existence of at least one nontrivial solution to the auxiliary problem, see Theorem 3.1, and by truncating functions \(M_{i}\), we obtain the existence of at least one nontrivial solution to problem (1.1), see Theorem 1.2.
Throughout this paper, we shall assume that F satisfies the following conditions:
- (\(\boldsymbol{F}_{1}\)):
-
\(F\in C^{1}(\Omega \times \mathbb{R}^{n},\mathbb{R})\) and \(F (x,0_{\mathbb{R}^{n}} )=0\).
- (\(\boldsymbol{F}_{2}\)):
-
For all \((i,j)\in \lbrace 1,2,\dots ,n \rbrace ^{2}\), there exist positive functions \(b_{i_{j}}\) such that
$$\begin{aligned}& \bigl\vert F_{\xi _{i}}(x,\xi _{1},\dots ,\xi _{n}) \bigr\vert \leq \sum_{j=1}^{n} b_{i_{j}}(x) \vert \xi _{j} \vert ^{r_{i_{j}}(x)-1}, \end{aligned}$$where \(1< r_{i_{j}}(x)<\inf_{x\in \Omega}h_{i}(x)\) for all \(x\in \Omega \). The weight-functions \(b_{ii}\) (resp. \(b_{i_{j}}\) if \(i\neq j\)) belong to the generalized Lebesgue spaces \(L^{\alpha _{ii}}(\Omega )\) (resp. \(L^{\alpha _{i_{j}} }(\Omega )\)), with
$$ \alpha _{ii}(x)=\frac{h_{i}(x)}{h_{i}(x)-1}, \qquad \alpha _{i_{j}} (x) = \frac{h_{i}^{\ast }(x)h_{j}^{\ast }(x)}{h_{i}^{\ast }(x)h_{j}^{\ast }(x)-h_{i}^{\ast }(x)-h_{j}^{\ast }(x)}. $$ - (\(\boldsymbol{F}_{3}\)):
-
There exist \(K>0\) and \(\gamma _{i} \in (h_{i}^{+}, \inf \{s_{i}^{-}, \ell _{i}^{-}\})\) such that for all \((x,\xi _{1},\dots ,\xi _{n})\in \Omega \times \mathbb{R}^{n}\) where \(| u_{i}|^{\gamma _{i}}\geq K \),
$$ 0 < F(x,\xi _{1},\dots ,\xi _{n})< \sum _{i=1}^{n} \frac{\xi _{i}}{\gamma _{i}} F_{\xi _{i}}(x,\xi _{1},\dots ,\xi _{n}). $$ - (\(\boldsymbol{F}_{4}\)):
-
There exists a positive constant c such that
$$ \bigl\vert F(x,\xi _{1},\dots ,\xi _{n}) \bigr\vert \leq c \Biggl(\sum_{i=1}^{n} \vert \xi _{i} \vert ^{w_{i}(x)} \Biggr), \quad \text{for all } (x,\xi _{1},\dots ,\xi _{n})\in \Omega \times \mathbb{R}^{n}, $$where \(w_{i} \in C_{+}(\overline{\Omega})\) and \(q_{i}^{+}< w_{i}^{-}\leq w_{i}^{+}< \inf \{s_{i}^{-}, \ell _{i}^{-} \} \leq \inf \{s_{i}^{+}, \ell _{i}^{+}\}\), for all \(1\leq i \leq n \).
Example 1.1
There are many potential functions F satisfying assumptions \((\boldsymbol{F}_{1})\) and \((\boldsymbol{F}_{2})\). For example, when \(n=2\), take \(F(x,u_{1},u_{2})=b_{12}(x)|u_{1}|^{r_{1}(x)}|u_{2}|^{r_{2}(x)} \), where \(\frac{r_{1}(x)}{h_{1}(x)}+\frac{r_{2}(x)}{h_{2}(x)}< 1\), and the positive weight-function \(b_{12} \in L^{\alpha (x)}(\Omega ) \) with
By the standard calculus, we can verify that F satisfies the assumption \((\boldsymbol{F}_{1})\). Moreover, by using Young inequality, we can check the assumption \((\boldsymbol{F}_{2})\).
The main result of our paper is the following.
Theorem 1.2
Assume that conditions \((\boldsymbol{A}_{1})\)–\(( \boldsymbol{A}_{4})\), \((\boldsymbol{M})\), and \((\boldsymbol{F}_{1})\)–\((\boldsymbol{F}_{4})\) hold. Then there exists \(\lambda ^{\star} >0\) such that for all \(\lambda \geq \lambda _{\star}\), problem (1.1) has at least one nontrivial solution in X. Moreover, if \(u_{\lambda}\) is a weak solution of problem (1.1), then \(\lim_{ \lambda \to +\infty } \| u_{\lambda}\|=0\).
The paper is organized as follows: In Sect. 2 we give some preliminary results of the variable exponent spaces. In Sect. 3 we introduce the auxiliary problem and obtain a nontrivial solution for the auxiliary problem. Section 4 is dedicated to proving the main results. Finally, in Sect. 5, we illustrate the degree of generality of the kind of problems we studied in this paper.
2 Preliminaries and basic notations
In this section, we introduce some definitions and results which will be used in the next section. Throughout our work, let Ω be a bounded domain of \(\mathbb{R}^{N}\) (\(N\geq 2\)) with a Lipschitz boundary ∂Ω, and let us denote by \(\mathcal{D}\) either Ω or its boundary ∂Ω. Denote
For all \(p \in \mathcal{M}_{+}(\mathcal{D})\), denote \(p^{+}:= \sup_{x\in \mathcal{D}}p(x)\) and \(p^{-}:= \inf_{x\in \mathcal{D}}p(x)\). Also for all \(p \in \mathcal{M}_{+}(\mathcal{D})\) and for a measure η on \(\mathcal{D}\), we define the variable exponent Lebesgue space as
where the functional \(\rho _{p,\mathcal{D}}:L^{p(x)}(\mathcal{D}) \to \mathbb{R}\) is defined as
The functional \(\rho _{p,\mathcal{D}}\) is called the \(p(x)\)-modular of the \(L^{p(x)}(\mathcal{D})\) space, it has played an important role in manipulating the generalized Lebesgue–Sobolev spaces. We endow the space \(L^{p(x)}(\mathcal{D})\) with the Luxemburg norm
Then \((L^{p(x)}(\mathcal{D}),\|u\|_{L^{p(x)}(\mathcal{D})} )\) is a separable and reflexive Banach space (see, e.g., Kováčik and Rákosník [34, Theorem 2.5, Corollary 2.7]). In the subsequent sections, the \(L^{p(x)}\)-spaces under consideration will be \(L^{p(x)}(\Omega ):=L^{p(x)}(\Omega ,dx)\), and \(L^{p(x)}(\partial \Omega ):=L^{p(x)}(\partial \Omega ,d\sigma )\) for an appropriate measure σ supported on ∂Ω. Let us now recall more basic properties concerning the Lebesgue spaces.
Proposition 2.1
(Kováčik and Rákosník [34, Theorem 2.8])
Let p and q be variable exponents in \(\mathcal{M}_{+}(\mathcal{D})\) such that \(p\leq q\) in \(\mathcal{D} \), where \(0<\operatorname{meas}(\mathcal{D})<\infty \). Then the embedding \(L^{q(x)}(\mathcal{D})\hookrightarrow L^{p(x)}(\mathcal{D})\) is continuous.
Furthermore, the following Hölder-type inequality:
holds for all \(u\in L^{p(x)}({\mathcal{D}})\) and \(v\in L^{p'(x)}({\mathcal{D}})\) (see, e.g., Kováčik and Rákosník [34, Theorem 2.1]), where we denoted by \(L^{p'(x)}(\mathcal{D})\) the topological dual space (or the conjugate space) of \(L^{p(x)}(\mathcal{D})\), obtained by conjugating the exponent pointwise, that is, \(1/p(x)+1/p'(x)=1\) (see, e.g., Kováčik and Rákosník [34, Corollary 2.7]). Moreover, if \(h_{1},h_{2},h_{3}:\mathcal{D}\to (1,\infty )\) are Lipschitz continuous functions such that
then for all \(u \in L^{h_{1}(x)}(\mathcal{D})\), \(v\in L^{h_{2}(x)}( \mathcal{D})\), \(w\in L^{h_{3}(x)}(\mathcal{D})\), the following inequality holds:
If \(u\in L^{p(x)}({\mathcal{D}})\) and \(p<\infty \), we have the following properties (see, for example, Fan and Zhao [23, Theorems 1.3 and 1.4]):
As a consequence, we have the equivalence of modular and norm convergence
Proposition 2.2
(Edmunds and Rakosnik [22])
Let h and ℓ be variable exponents in \(\mathcal{M}_{+}(\mathcal{D})\) with \(1\leq h(x), \ell (x)\leq \infty \) a.e. x in Ω and \(p\in L^{\infty }(\Omega )\). Then if \(u\in L^{\ell (x)}(\Omega )\), \(u\neq 0\), it follows that
When \(h(x)=h\) is constant, we obtain \(\| | u | ^{h}\| _{L^{\ell (x)}(\Omega )} = \|u \|_{L^{h\ell (x)}(\Omega )}^{h} \).
Now, let us pass to the Sobolev space with variable exponent, that is,
where \(\partial _{x_{i}}u= \frac{\partial u}{\partial x_{i}}\) represent the partial derivatives of u with respect to \(x_{i}\) in the weak sense. This space has a corresponding modular given by
which yields the norm
Another possible choice of norm in \(W^{1,p(x)}(\Omega )\) is \(\|u\| _{L^{p(x)}(\Omega )}+\| \nabla u \| _{L^{p(x)}(\Omega )}\). Both norms turn out to be equivalent, but we use the first one for convenience. It is well known that \(W^{1,p(x)}(\Omega )\) is a separable and reflexive Banach spaces (see, e.g., Kováčik and Rákosník [34, Theorem 3.1]). As usual, we define by \(p^{\ast }(x)\) the critical Sobolev exponent and \(p^{\partial }(x)\) the critical Sobolev trace exponent, respectively, by
We recall the following crucial embeddings on \(W^{1,p(x)}(\Omega )\).
Proposition 2.3
(Diening et al. [21], Edmunds and Rakosnik [22])
Let p be Lipschitz continuous and satisfying \(1< p^{-}\leq p(x)\leq p^{+}< N\), and let \(q\in C(\overline{\Omega})\) satisfy \(1\leq q(x) \leq p^{\ast}(x)\), for all \(x\in \overline{\Omega}\). Then there exists a continuous embedding \(W^{1,p(x)}(\Omega )\hookrightarrow L^{q(x)}(\Omega )\). If we assume in addition that \(1\leq q(x)< p^{\ast}(x)\) for all \(x\in \overline{\Omega}\), then this embedding is compact.
Proposition 2.4
(Diening et al. [21], Edmunds and Rakosnik [22])
Let \(p\in W^{1,h}(\Omega )\) with \(1\leq p_{-}\leq p_{+}< N< h\). Then for all \(q\in C(\partial \Omega )\) satisfying \(1\leq q(x) \leq p^{\partial}(x)\) for \(x\in \partial \Omega \), there is a continuous boundary trace embedding \(W^{1,p(x)}(\Omega )\hookrightarrow L^{q(x)}(\partial \Omega )\). If we assume in addition that \(1\leq q(x)< p^{\ast}(x)\) for all \(x\in \overline{\Omega}\), then this embedding is compact.
For detailed properties of the variable exponent Lebesgue–Sobolev spaces, we refer the reader to Diening et al. [21], Kováčik and Rákosník [34]. As is well known, the use of critical point theory needs the well-known Palais–Smale condition (\((PS)_{c}\), for short), which plays a central role.
Definition 2.5
Consider a function \(E:X\to \mathbb{R}\) of class \(C^{1}\), where X is a real Banach space. We say that a sequence \(\{u_{m}\}\) is a Palais–Smale sequence for the functional E if
We say that \(\{u_{m}\}\) is a Palais–Smale sequence with energy level c (or \((u_{m})\) is \((PS)_{c}\), for short). Moreover, if every \((PS)_{c}\) sequence for E has a strongly convergent subsequence in X, then we say that E satisfies the Palais–Smale condition at level c (or E is \((PS)_{c}\), short).
Our main tool is the following classical Mountain Pass Theorem.
Theorem 2.6
(Rabinowitz [43])
Let X be a real infinite-dimensional Banach space and let \(E: X \to \mathbb{R}\) be of class \(C^{1}\) and satisfying the \((PS)_{c}\) such that \(E(0_{X})=0\). Assume that
- (\(H_{1}\)):
-
there exist positive constants ρ, \(\mathcal{R} \) such that \(E(u)\geq \mathcal{R}\), for all \(u\in \partial B_{\rho}\cap X\);
- (\(H_{2}\)):
-
there exists \(z\in X\) with \(\Vert z \Vert _{X} >\rho \) such that \(E(z)<0\).
Then E has a critical value \(c\geq \mathcal{R}\), which can be characterized as
where
In the sequel, we shall use the product space \(X:=\prod_{i=1}^{n} (W^{1,h_{i}(x)}(\Omega )\cap W^{1,p_{i}(x)}( \Omega ) ) \), equipped with the norm \(\| u\|:=\max_{1\leq i \leq n} \{ \| u_{i}\|_{i} \} \), for all \(u=(u_{1},u_{2},\dots ,u_{n})\in X \), where \(\| u_{i}\|_{i}:= \| u_{i}\|_{1,p_{i}(x)}+ \mathcal{K}(k_{i}^{3})\| u_{i} \|_{1,q_{i}(x)}\) is the norm of \(W^{1,h_{i}(x)}(\Omega )\cap W^{1,p_{i}(x)}(\Omega ) \), for all \(i \in \{1,2,\dots ,n\}\).
Definition 2.7
We say that \(u=(u_{1},u_{2},\dots ,u_{n}) \in X\) is a weak solution of the system (1.1) if
for all \(v=(v_{1},v_{2},\dots ,v_{n})\in X= \prod_{i=1}^{n}(W^{1,h_{i}(x)}( \Omega )\cap W^{1,p_{i}(x)}(\Omega ))\).
The energy functional \(E_{\lambda}:X\rightarrow \mathbb{R} \) associated with problem (1.1) is defined as \(E_{\lambda}(\cdot ):= \Phi (\cdot )- \Psi (\cdot ) -\Upsilon (\cdot )- \mathcal{F}_{\lambda}(\cdot )\), where \(\Phi , \Psi , \text{ and } \mathcal{F}_{\lambda}:X\rightarrow \mathbb{R} \) are given by
for all \(u=(u_{1},\dots ,u_{n})\) in X, where \(\widehat{M}_{i}(\tau )= \int _{0}^{\tau}M_{i}(s)\,ds \). By standard calculus, one can see that, under the above assumptions, the energy functional \(E_{\lambda}:X\to \mathbb{R}^{N}\) corresponding to problem (1.1) is well defined and \(E_{\lambda}\in C^{1}(X,\mathbb{R})\) with
for all \(v=(v_{1},v_{2},\dots ,v_{n})\in X\). So the critical points of functional \(E_{\lambda}\) are weak solutions of system (1.1).
To prove our existence result, since we have lost compactness in the inclusions \(W^{1,h_{i}(x)}(\Omega )\hookrightarrow L^{h_{i}^{\ast}(x)}(\Omega )\) and \(W^{1,h_{i}(x)}(\Omega )\hookrightarrow L^{h_{i}^{\partial}(x)}( \partial \Omega )\), for all i in \(\{1,2,\dots ,n\}\), we can no longer expect the Palais–Smale condition to hold. Nevertheless, we can prove a local Palais–Smale condition that will hold for the energy functional \(E_{\lambda}\) below a certain value of energy, by using the principle of concentration–compactness for the variable exponent Sobolev space \(W^{1,h_{i}(x)}(\Omega )\). For the reader’s convenience, we state this result in order to prove Theorem 1.2, see Bonder et al. [6, 7] for its proof.
Now, let \(\mathcal{O}\) be a different subset of ∂Ω, a closed set (possibly empty). Set
where closure is taken with respect to \(\| v \| _{1,h(x)}\). This is the subspace of functions vanishing on \(\mathcal{O}\). Evidently, \(W^{1,h(x)}_{\emptyset}(\Omega ) =W^{1,h(x)}(\Omega )\). In general, \(W^{1,h(x)}_{\mathcal{O}}(\Omega )=W^{1,h_{i}(x)}(\Omega )\) if and only if the \(h_{i}(x)\)-capacity of \(\mathcal{O}\) equals zero; for more details, we refer the interested readers to Harjulehto et al. [29]. The best Sobolev trace constant \(T_{i}(h_{i}(x),\ell _{i}(x),\mathcal{O})\) is defined by
Theorem 2.8
(Bonder et al. [6])
Let \(h_{i}\in C_{+}(\overline{\Omega})\), \(\ell _{i} \in C_{+}(\partial \Omega )\) be such that \(\ell _{i}(x)\leq h_{i}^{ \partial }(x)\), for all \(x \in \partial \Omega \), and \(\{u_{m}\}_{m\in \mathbb{N}}\) be a sequence in \(W^{1,h_{i}(x)}(\Omega )\) such that \(u_{i_{m}} \rightharpoonup u_{i}\) weakly in \(W^{1,h_{i}(x)}(\Omega )\). Then there exist a countable index set \(J_{i}^{1}\), positive numbers \(\{\mu _{i_{j}}\}_{j \in J_{i}^{1}}\) and \(\{\nu _{i_{j}}\}_{j \in J_{i}^{1}}\), and \(\{x_{j}\}_{j\in J_{i}^{1}}\subset \mathcal{C}_{h_{i}}^{1}= \{x\in \partial \Omega \colon \ell _{i}(x)=h_{i}^{\partial}(x)\}\) such that
where
is the localized Sobolev trace constant with \(\Omega _{\epsilon ,j} =\Omega \cap B_{\epsilon}(x_{j})\) and \(\Lambda _{\epsilon ,j} =\Omega \cap \partial B_{\epsilon}(x_{j})\).
Theorem 2.9
(Bonder and Silva [7])
Let \(h_{i}\) and \(s_{i}\) be variable exponents \(\in C_{+}(\overline{\Omega})\) such that \(s_{i}(x)\leq h_{i}^{ \ast }(x) \), for all \(x \in \overline{\Omega}\), and \(\{u_{m}\}_{m\in \mathbb{N}}\) be a sequence in \(W^{1,h_{i}(x)}(\Omega )\) such that \(u_{i_{m}} \rightharpoonup u_{i}\) weakly in \(W^{1,h_{i}(x)}(\Omega )\). Then there exist a countable set \(J_{i}^{2}\), positive numbers \(\{\mu _{i_{j}}\}_{j \in J_{i}^{2}}\) and \(\{\nu _{i_{j}}\}_{j \in J_{i}^{2}}\), and \(\{x_{j}\}_{j\in J_{i}^{2}}\subset \mathcal{C}_{h_{i}}^{2}= \{x\in \Omega \colon s_{i}(x)=h_{i}^{\ast}(x)\}\) such that
where
is the best constant in the Gagliardo–Nirenberg–Sobolev inequality for variable exponents.
Notations. Weak (resp. strong) convergence will be denoted by ⇀ (resp., →), \(C_{i}\), \(C_{i_{j}}\), \(c_{j}\), and \(c_{i_{j}}\) will denote positive constants which may vary from line to line and can be determined in concrete conditions. Here, \(X^{\ast}\) denotes the dual space of X, \(\delta _{x_{j}}\) is the Dirac mass at \(x_{j}\), for all \(\rho >0\), \(x\in \Omega \), where \(B(x,\rho )\) denotes the ball of radius ρ centered at x.
3 The auxiliary problem and variational framework
In order to prove Theorem 1.2, we shall introduce the auxiliary problem by defining the auxiliary functional \(E_{\theta ,\lambda}\) and showing that the energy functional \(E_{\theta ,\lambda}\) has the geometry of the Mountain Pass Theorem 2.6.
By assumption \((\boldsymbol{M})\), we see that the functions \(M_{i}\) are bounded only from below and do not give us enough information about the behavior of \(M_{i}\) at infinity, which makes it difficult to prove that the functional \(E_{\lambda}\) has the geometry of the Mountain Pass Theorem and that the sequence of Palais–Smale is bounded in X. Hence, we truncate functions \(M_{i}\) and study the associated truncated problem.
Take \(\gamma _{i}\) as in assumption (\(\boldsymbol{F}_{3}\)) and \(\theta _{i} \in \mathbb{R}\), for all \(i\in \{1,\dots ,N\}\) such that \(\mathfrak{M}_{i}^{0} < \theta _{i} < \frac{\gamma _{i}\mathfrak{M}_{i}^{0}}{p_{i}^{+}\max \{\beta _{1_{i}},\beta _{2_{i}}\}}\). By assumption \((\boldsymbol{M})\), there exists \(\tau _{i}^{0}>0\) such that \(M_{i}(\tau _{i}^{0})=\theta _{i}\). Thus, by setting
we can introduce the following auxiliary problem:
for \(1\leq i\leq n\), where \(\mathcal{A}_{i}\), \(\mathcal{B}_{i}\), \(F_{u_{i}}\), and λ are as in Sect. 1. By assumption \((\boldsymbol{M})\), we also know that
Now, the next step is to prove that the auxiliary problem (3.1) has a nontrivial weak solution. We obtain the following result.
Theorem 3.1
Suppose that conditions \((\boldsymbol{A}_{1})\)–\(( \boldsymbol{A}_{4})\), \((\boldsymbol{M})\), and \((\boldsymbol{F}_{1})\)–\((\boldsymbol{F}_{4})\) hold. Then there exists a constant \(\lambda _{\star} >0\), such that if \(\lambda \geq \lambda _{\star}\), then problem (3.1) has at least one nontrivial solution in X.
For the proof of Theorem 3.1, we shall need some technical results. We observe that the auxiliary problem (3.1) has a variational structure; indeed, it is the Euler–Lagrange equation of the functional \(E_{\theta ,\lambda }: X\to \mathbb{R}\) defined as follows:
where \(\widehat{M}_{\theta _{i}}(\tau )= \int _{0}^{\tau}M_{\theta _{i}}(s)\,ds \). Moreover, the functional \(E_{\theta ,\lambda }\) is Fréchet differentiable in \(u\in X\) and, for all \(v=(v_{1},\dots ,v_{n})\in X\),
Now we prove that the functional \(E_{\theta ,\lambda }\) has the geometric features required by the Mountain Pass Theorem 2.6.
Lemma 3.2
Suppose that conditions \((\boldsymbol{A}_{1})\)–\((\boldsymbol{A}_{4})\), \((\boldsymbol{M})\) and \((\boldsymbol{F}_{1})\)–\(( \boldsymbol{F}_{4})\) hold. Then there exist positive constants \(\mathcal{R}\) and ρ such that \(E_{\theta ,\lambda }(u)\geq \mathcal{R}>0\), for all \(u\in X\) with \(\Vert u \Vert =\rho \).
Proof
For all \(u=(u_{1},\dots ,u_{n})\in X\), we obtain, under the assumptions \((\boldsymbol{A}_{2})\) and \((\boldsymbol{A}_{4})\),
Hence, by using assumption \((\boldsymbol{F}_{4})\), and relations (2.2)–(2.3), we have
By the Sobolev Embedding Theorem and taking \(0< \Vert u \Vert = \max_{1\leq i\leq n} \{\| u_{i}\|_{1,p_{i}(x)}+ \mathcal{K}(k_{i}^{3}) \| u_{i}\|_{1,q_{i}(x)} \}= \rho <1\), there exist positive constants \(c_{1_{i}}\), \(c_{2_{i}}\), \(c_{3i}\), \(c_{4i}\), and \(c_{5i}\) such that
Since \(q_{i}^{+}< w_{i}^{- }< \inf \{s_{i}^{-},\ell _{i}^{-}\}\), there exists a positive constant \(\mathcal{R}\) such that \(E_{\theta ,\lambda }(u)\geq \mathcal{R}>0\), with \(\Vert u \Vert =\rho \). □
Lemma 3.3
Suppose that conditions \((\boldsymbol{A}_{1})\)–\((\boldsymbol{A}_{4})\), \((\boldsymbol{M})\), and \((\boldsymbol{F}_{4})\) hold. Then for every positive function λ, there exists a nonnegative function \(e\in X\), independent of λ, such that \(\Vert z \Vert >R\) and \(E_{\theta ,\lambda }(z)<0\).
Proof
By the assumptions \((\boldsymbol{A}_{1})\) and \((\boldsymbol{A}_{4})\), for all \(\delta >0\) and \(u\in X\), we have
By this inequality and assumption \((\boldsymbol{F}_{4})\), we have, for \(e=(e_{1},\dots ,e_{n}) \in X \setminus \{(0,\dots ,0) \}\) and each \(\delta >1\),
which tends to −∞ as \(\delta \to +\infty \) since \(\min \{s_{i}^{-},\ell _{i}^{-}\} >w_{i}^{-}>q_{i}^{+}\). So, the lemma is proven by choosing \(z=\delta _{*} e\) with \(\delta _{*} >0\) sufficiently large. □
Now, by the Mountain Pass Theorem 2.6 without the Palais–Smale condition, we get a sequence \(\{u_{m}\}_{m\in \mathbb{N}}\subset X\) such that \(E_{\theta ,\lambda}(u_{m}) \to c_{\theta ,\lambda} \) and \(E_{ \theta ,\lambda}'(u_{m}) \to 0\), where \(c_{\theta ,\lambda} := \inf_{\phi \in \Gamma} \max_{ \delta \in [ 0,1 ] }E_{\lambda}(\phi (\delta )) \), and
Lemma 3.4
Suppose that conditions \((\boldsymbol{A}_{1})\)–\((\boldsymbol{A}_{2})\), \((\boldsymbol{M})\), and \((\boldsymbol{F}_{4})\) hold. Then \(\lim_{\lambda \to +\infty } c_{\theta ,\lambda}=0\).
Proof
Let \(z=(z_{1},\dots ,z_{n})\in X\) be the function given by Lemma 3.3. Then \(\lim_{\delta \to \infty }E_{\theta ,\lambda }(\delta z)=- \infty \), for each \(\lambda >0\), so it follows that there exists \(\delta _{\lambda}>0\) such that \(E_{\theta ,\lambda}(\delta _{\lambda}z)=\max_{\delta \geq 0}E_{ \theta ,\lambda}(\delta z)\). Hence, \(\langle E'_{\theta ,\lambda}(\delta _{\lambda}z), \delta _{\lambda}z \rangle =0 \), so it follows by relation (3.3) that
By construction, \(z_{i}\geq 0\) a.e. in Ω, for all \(i\in \{1,2,\dots ,n\}\). Therefore, by assumption \((\boldsymbol{F}_{3})\) and relation (3.4), we get
On the other hand, by assumption \((\boldsymbol{A}_{2})\) and inequalities (2.2), (2.3), we get
Therefore, from relations (3.5), (3.6), and inequalities (2.2), (2.3), we obtain
Next, we shall show that the sequence \(\{\delta _{\lambda }\}\) is bounded in \(\mathbb{R}\). Indeed, we suppose by contradiction that \(\{\delta _{\lambda }\}\) is unbounded. Then there is a subsequence denoted by \(\{\delta _{\lambda _{m}}\}\), with \(\delta _{\lambda _{m}} \to \infty \), as \(m\to +\infty \). Then by relation (3.7),
where \(q_{M}^{+}= \max_{1\leq i\leq n}\{ q_{i}^{+}\}\). Therefore, when taking the limit as \(m\to +\infty \), we get a contradiction because \(p_{i}^{+}< q_{M}^{+}<\inf \{s_{i}^{-},\ell _{i}^{-}\}\). Thus, we can conclude that \(\{\delta _{\lambda }\}\) is indeed bounded in \(\mathbb{R}\).
Consider a sequence \(\{\lambda _{m}\}_{m\in \mathbb{N}}\) such that \(\lambda _{m}\to +\infty \) and let \(\delta _{0}\geq 0\) be such that \(\delta _{\lambda _{m}} \to \delta _{0}\), as \(m\to +\infty \). Then by continuity of \(M_{\theta _{i}}\), \(\{M_{\theta _{i}}(\mathcal{A}_{i}(\delta _{\lambda _{m}}z_{i}))\}_{m \in \mathbb{N}}\) is bounded, for all \(i\in \{1,2,\dots ,n\}\). Therefore, there exists \(C>0\) such that
so by inequality (3.5), we have
We shall prove that \(\delta _{0}=0\). Indeed, if \(\delta _{0}>0\), then by assumption \((\boldsymbol{F}_{2})\), there exist positive functions \(b_{i_{j}}\) (\(1\leq i,j \leq n\)), such that
Thus, by the Lebesgue Dominated Convergence Theorem, we get
By remembering that \(\lambda _{m}\to +\infty \), we find
This contradicts the fact (3.9), so we can deduce that \(\delta _{0}=0\).
Next, we consider the following path \(\phi _{*}(\delta )=\delta z\) for \(\delta \in [0,1]\) which belongs to Γ. By using assumption \((\boldsymbol{F}_{3})\), we obtain
On the other hand, since \(M_{\theta _{i}}\) are continuous for all \(1\leq i \leq n\), and \(\delta _{0}=0\), we get \(\lim_{ m\to +\infty } \widehat{M_{\theta _{i}}} (\mathcal{A}_{i}( \delta _{\lambda _{m}}z_{i}) )=0 \), for all \(i\in \{1,2,\dots ,n\}\). Thus, from relation (3.10), we get \(\lim_{ m\to +\infty } c_{\theta , \lambda _{m}}=0\). Moreover, by using also assumption \((\boldsymbol{F}_{3})\), we verify that the sequence \(\{c_{\theta , \lambda}\}_{\lambda}\) is monotone. Therefore, we have completed the proof. □
Lemma 3.5
If \(\lbrace u_{m}=(u_{1_{m}},u_{2_{m}},\dots ,u_{n_{m}}) \rbrace _{m\in \mathbb{N}} \) is a Palais–Smale sequence for \(E_{\theta ,\lambda }\), then \(\lbrace u_{m} \rbrace _{m}\) is bounded in X.
Proof
Let \(\lbrace u_{m}=(u_{1_{m}},u_{2_{m}},\dots ,u_{n_{m}}) \rbrace _{m} \) be a \((PS)_{c}\) for \(E_{\theta ,\lambda }\). Then we have
On the other hand, for all \(v=(v_{1},v_{2},\dots ,v_{n})\in X\), we have
Thus,
where \(\frac{u_{m}}{\gamma}=(\frac{u_{1_{m}}}{\gamma _{1}}, \frac{u_{2_{m}}}{\gamma _{2}},\dots ,\frac{u_{n_{m}}}{\gamma _{n}})\). Therefore, by using assumptions \((\boldsymbol{A}_{4})\), \((\boldsymbol{M})\), and \((\boldsymbol{F}_{3})\), we can conclude that
By using assumption \((\boldsymbol{A}_{4})\), we can find positive constants \(C_{i1}\) and \(C_{i2}\) such that
To prove the assertion, we assume by contradiction that \(\Vert u_{i_{m}} \Vert _{i} =\| u_{i}\|_{1,p_{i}(x)}+ \mathcal{K}(k_{i}^{3}) \| u_{i}\|_{1,q_{i}(x)} \to + \infty \). So, if \(k_{i}^{3}=0\), then by using relation (2.2), we have \(E_{\theta ,\lambda }(u_{m})-\langle E_{\theta ,\lambda }'(u_{m}), \frac{u_{m}}{\gamma }\rangle \geq \sum_{i=1}^{n} C_{i} \| u_{i_{m}} \|_{i}^{p_{i}^{-}} \). Thus, we can find \(c+ o_{m}(1) \geq \sum_{i=1}^{n} C_{i} \| u_{i_{m}}\|_{i}^{p_{i}^{-}} \). Since \(p_{i}^{-}>1\), we obtain a contradiction. Hence, we deduce that \(\lbrace u_{m} \rbrace \) is bounded in X.
When \(k_{i}^{3}>0\), we have three cases to analyze:
-
(i)
\(\Vert u_{i_{m}} \Vert _{1,p_{i}(x)} \to +\infty \) and \(\Vert u_{i_{m}} \Vert _{1,q_{i}(x)} \to +\infty \), as \(m \to +\infty \),
-
(ii)
\(\Vert u_{i_{m}} \Vert _{1,p_{i}(x)} \to +\infty \) and \(\Vert u_{i_{m}} \Vert _{1,q_{i}(x)}\) is bounded,
-
(iii)
\(\Vert u_{i_{m}} \Vert _{1,p_{i}(x)}\) is bounded and \(\Vert u_{i_{m}} \Vert _{1,q_{i}(x)}\to +\infty \).
In the case (i), for m sufficiently large, we have \(\| u_{i_{m}}\|_{1,q_{i}(x)}^{q^{-}} \geq \| u_{i_{m}}\|_{1,q_{i}(x)}^{p^{-}} \). Hence, by inequality (3.12), we get
and this is a contradiction. In the case (ii), by using inequality (3.12), we conclude that
Hence, we also get a contradiction when limit as \(m\to +\infty \) because \(p_{i}^{-}>1\). In the last case (iii), the proof is similar as in the case (ii), so we shall omit it. Finally, we can deduce that \(\{u_{m}\} \) is a bounded sequence in X. □
Next, we shall prove that the auxiliary problem (3.1) possesses at least one nontrivial weak solution.
Proof of Theorem 3.1
By Lemmas 3.2 and 3.3, the functional \(E_{\theta , \lambda}\) satisfies the geometric structure required by the Mountain Pass Theorem 2.6. Now, it remains to check the validity of the Palais–Smale condition. Let \(\{u_{m}=(u_{1_{m}},u_{2m},\dots ,u_{n_{m}}) \}_{m\in \mathbb{N}}\) be a Palais–Smale sequence at the level \(c_{\theta , \lambda}\) in X. Then Lemma 3.4 implies that there exists \(\lambda _{*}\) such that
where \(\overline{T_{i}}_{x_{j}}\) and \(S_{i } \) are respectively given by relations (2.10) and (2.14), and
So, there exists a strongly convergent in X subsequence. Indeed, applying Lemma 3.5, \(\{ u_{m}\}_{m\in \mathbb{N}}\) is bounded in X, so, passing to a weakly convergent in X subsequence, still denoted by \(\{ u_{m}\}_{m}\), there exist positive bounded measures \(\mu _{i}\), \(\nu _{i} \in \Omega \) and \(\overline{\nu}_{i} \in \partial \Omega \) such that \(|\nabla u_{i_{m}}|^{h_{i}(x)}\rightharpoonup \mu _{i} \), \(| u_{i_{m}}|^{s_{i}(x)}\rightharpoonup \nu _{i} \), and \(| u_{i_{m}}|^{ \ell _{i}(x)}\rightharpoonup \overline{\nu _{i}} \). Hence, by Theorems 2.8 and 2.9, if \(\bigcup_{i=1}^{n}(J_{i}^{1}\cup J_{i}^{2}) =\emptyset \), then \(u_{i_{m}} \rightharpoonup u_{i}\) in \(L^{s_{i}(x)}(\Omega )\) and \(u_{i_{m}} \rightharpoonup u_{i}\) in \(L^{\ell _{i}(x)}(\partial \Omega )\), for all \(1\leq i\leq n\). Let us prove that if
and \(\{u_{m}\}_{m\in \mathbb{N}} \) is a Palais–Smale sequence with energy level \(c_{\theta ,\lambda}\), then \(J_{i}^{1}\cup J_{i}^{2}=\emptyset \), for all \(1\leq i\leq n\). In fact, suppose there is an \(i\in \{1,\dots ,n\} \) such that \(J_{i}^{1}\cup J_{i}^{2}\) is nonempty, then \(J_{i}^{1}\neq \emptyset \) or \(J_{i}^{2}\neq \emptyset \).
First, we consider the case \(J_{i}^{1}\neq \emptyset \). Let \(x_{j}\in \mathcal{C}^{1}_{h_{i}}\) be a singular point of the measures \(\mu _{i}\) and \(\overline{\nu _{i}}\). Consider \(\psi \in C^{\infty}_{0}(\mathbb{R}^{N})\) such that \(0\leq \psi (x) \leq 1\), \(\psi (0)=1\), and \(\operatorname{supp} \psi \subset B(0,1)\). We consider, for each \(j\in J_{i}^{1}\) and any \(\varepsilon >0\), the functions \(\psi _{j,\varepsilon} := \psi (\frac{x-x_{j}}{\varepsilon} )\), for all \(x \in \mathbb{R}^{N}\). Notice that \(\psi _{j,\varepsilon} \in C^{\infty}_{0}(\mathbb{R}^{N}, [ 0,1 ] )\), \(|\nabla \psi _{j,\epsilon}|_{\infty}\leq \frac{2}{\varepsilon}\), and
Since \(\{u_{i_{m}}\}_{m}\) is bounded in \(W^{1,h_{i}(x)}(\Omega )\cap W^{1,p_{i}(x)}(\Omega )\), the sequence \(\lbrace u_{i_{m}}\psi _{j,\varepsilon} \rbrace \) is also bounded in \(W^{1,h_{i}(x)}(\Omega )\cap W^{1,p_{i}(x)}(\Omega )\). So, by relation (3.11), we obtain \(\langle E_{\theta ,\lambda }'(u_{1_{m}},\dots , u_{i_{m}},\dots , u_{n_{m}}), (0, \dots ,u_{i_{m}}\psi _{j,\varepsilon},\dots ,0 )\rangle \rightarrow 0 \) as \(m \rightarrow +\infty \). Therefore, we have, as \(m \rightarrow +\infty \),
and so
Next, we shall prove that
Notice that, due to assumption \((\boldsymbol{A}_{2})\), it suffices to show that
and
First, by applying Hölder inequality, we obtain
where, since \(\lbrace u_{i_{m}} \rbrace \) is bounded, the real-valued sequence \(\Vert \vert \nabla u_{i_{m}} \vert ^{p_{i}(x)-1} \Vert _{L^{ \frac{p_{i}(x)}{p_{i}(x)-1}}(\Omega )}\) is also bounded, thus there exists a positive constant \(C_{i}\) such that
Moreover, the sequence \(\{ u_{i_{m}}\}\) is bounded in \(W^{1,p_{i}(x)}(B(x_{j},2\varepsilon ))\), so there is a subsequence, again denoted by \(\lbrace u_{i_{m}} \rbrace \), converging weakly to \(u_{i}\) in \(L^{p_{i}(x)}(B(x_{j},2\varepsilon ))\). Therefore,
Note that
where \(\omega _{N}\) is the surface area of the N-dimensional unit sphere. Since
we obtain that \(\Vert \nabla \psi _{j,\varepsilon} u_{i} \Vert _{L^{p_{i}(x)}( \Omega )}\to 0\), which implies
Since the sequence \(\lbrace u_{i_{m}} \rbrace \) is bounded in \(W^{1,h_{i}(x)}(\Omega )\cap W^{1,p_{i}(x)}(\Omega ) \), we may assume that \(\mathcal{A}_{i}(u_{i_{m}}) \to \xi _{i}\geq 0 \), as \(m \to +\infty \). Note that \(M_{i}(\xi _{i})\) is is continuous, so we have \(M_{i} (\mathcal{A}_{i}(u_{i_{m}}) ) \to M_{i}(\xi _{i})\geq \mathfrak{M}_{i}^{0}>0 \), as \(m \to +\infty \). Therefore, by relation (3.17), we obtain
Analogously, we can verify relation (3.16). Hence, we have completed the proof of relation (3.14). Similarly, we can also obtain
By applying Hölder inequality, assumption \((\boldsymbol{F}_{2})\), and the fact that \(0\leq \psi _{j,\varepsilon} \leq 1\), we have
This yields
and the last term on the right-hand side goes to zero, because
Therefore, we have completed the proof of relation (3.19). On the other hand, we have
and since \(\mathcal{C}_{h_{i}}^{1} \cap \mathcal{C}_{h_{i}}^{2}=\emptyset \), for \(\epsilon >0\) sufficiently small, we have
hence, when \(\epsilon \to 0\),
The function \(\psi _{j,\varepsilon}\) has compact support, so letting \(m \to +\infty \) and \(\varepsilon \to 0\) in relation (3.13), we get from relations (3.14)–(3.19),
Note that, when \(k_{i}^{3}=0\), we have \(h_{i}(x)=p_{i}(x)\). Hence, by using relation (2.8), we have
By applying the Lebesgue Dominated Convergence Theorem, we get
Therefore,
On the other hand, if \(k_{i}^{3}>0\), then \(h_{i}(x)=q_{i}(x)\) Therefore, it follows from relations (2.10) and (3.20) that
By applying the Lebesgue Dominated Convergence Theorem again, we get
Hence,
By combining relations (3.21) and (3.22), we have \(\mathfrak{M}_{i}^{0} ((1-\mathcal{K}(k_{i}^{3}))\min \{k_{1_{i}}^{0},k_{2_{i}}^{0} \}+ \mathcal{K}(k_{i}^{3}) \min \{k_{1_{i}}^{2},k_{2_{i}}^{2} \} ) \mu _{i_{j}} \leq \overline{\nu}_{i_{j}}\). By using relation (2.10), we obtain
which implies that \(\overline{\nu}_{i_{j}}=0\) or \(\overline{\nu}_{i_{j}}\geq \overline{T_{i}}_{x_{j}}^{N} ( \mathfrak{M}_{i}^{0} (\min \{k_{1_{i}}^{0},k_{2_{i}}^{0} \}(1- \mathcal{K}(k_{i}^{3})+\mathcal{K}(k_{i}^{3})\min \{k_{1_{i}}^{2}, k_{2_{i}}^{2} \}) ) )^{N/h_{i}(x_{j})}\), for all \(j \in J_{i}^{1}\). On the other hand, by using assumptions \((\boldsymbol{M})\) and \((\boldsymbol{F}_{3})\), we have
Hence, we have \(c_{\theta ,\lambda} \geq \sum_{i=1}^{n} ( \frac{ 1}{\gamma _{i}}-\frac{ 1}{\ell _{i}^{-}} ) \int _{\partial \Omega }|u_{i_{m}}|^{\ell _{i}(x)}\,d\sigma _{x} + o_{m}(1)\). Setting
as \(m\to +\infty \), we obtain
where \(D_{i}= \mathfrak{M}_{i}^{0} (\min \{k_{1_{i}}^{0}, k_{2_{i}}^{0} \}(1-\mathcal{K}(k_{i}^{3})+\mathcal{K}(k_{i}^{3})\min \{k_{1_{i}}^{2},k_{2_{i}}^{2} \}) )\). Since \(\kappa > 0\) is arbitrary and \(\ell _{i}\) are continuous functions for all \(i\in \{1,2,\dots ,n\}\), we get
Suppose that \(\bigcup_{i=1}^{n}J_{i}^{1}\neq \emptyset \), then
Therefore, if \(c_{\theta , \lambda} < \min_{1\leq i\leq n} \lbrace (\frac{1}{\gamma _{i}}- \frac{1}{{\ell _{i}}_{\mathcal{C}^{1}_{h_{i}}}^{-}} )\inf_{j\in J_{i}^{1}} \{\overline{T_{i}}_{x_{j}}^{N} (D_{i} )^{N/h_{i}(x_{j})} \} \rbrace \), the set \(\bigcup_{i=1}^{n}J_{i}^{1}\) is empty, which means that for all \(1\leq i\leq n\), \(\|u_{i_{m}}\|_{L^{\ell _{i}(x)}(\partial \Omega )} \to \|u_{i}\|_{L^{ \ell _{i}(x)}(\partial \Omega )}\). Since \(u_{m}\rightharpoonup u\) in X, we have for all \(i \in \lbrace 1,2,\dots ,n \rbrace \) that \(u_{i_{m}} \to u_{i}\) strongly in \(L^{\ell _{i}(x)}(\partial \Omega )\). Next, consider \(J_{i}^{2}\neq \emptyset \), by the same approach for the case \(J_{i}^{1}\). We have
where \(D_{i}= \mathfrak{M}_{i}^{0} (\min \{k_{1_{i}}^{0}, k_{2_{i}}^{0} \}(1-\mathcal{K}(k_{i}^{3})+\mathcal{K}(k_{i}^{3})\min \{k_{1_{i}}^{2},k_{2_{i}}^{2} \}) )\). Hence, we deduce that \(\bigcup_{i=1}^{n}J_{i}^{2}= \emptyset \), which means that for all \(1\leq i\leq n\), \(\|u_{i_{m}}\|_{L^{s_{i}(x)}(\Omega )} \to \|u_{i}\|_{L^{s_{i}(x)}( \Omega )}\). Since \(u_{m}\rightharpoonup u\) in X, we have \(u_{i_{m}} \to u_{i}\) strongly in \(L^{s_{i}(x)}(\Omega )\), for all \(i \in \lbrace 1,2,\dots ,n \rbrace \). On the other hand, we have
thus \(E_{\theta ,\lambda}'(u_{1_{m}},\dots ,u_{n_{m}})\to 0\), i.e., \(E_{\theta ,\lambda} '(u_{1_{m}},\dots ,u_{n_{m}})\) is a Cauchy sequence in \(X^{\ast}\). Furthermore, by using Hölder inequality again, we find
Similarly, we also have
Since \(\lbrace u_{1_{m}} \rbrace \) is a Cauchy sequence in \(L^{s_{1}(x)}(\Omega )\) and \(L^{\ell _{1}(x)}(\partial \Omega )\), it follows that \(\Psi '(u_{1_{m}},\dots ,u_{n_{m}})\) and \(\Upsilon '(u_{1_{m}},\dots ,u_{n_{m}})\) are Cauchy sequences in \(X^{\star}\). Moreover, by compactness of \(\mathcal{F}_{\lambda} '\), we have
which means that \(\mathcal{F}_{\lambda} '(u_{1_{m}},\dots ,u_{n_{m}})\) is a Cauchy sequence also in \(X^{\ast}\). Therefore, invoking some elementary inequalities (see, e.g., Hurtado et al. [32, Auxiliary Results]), we conclude that for all \(\varrho , \zeta \in \mathbb{R}^{N} \),
where ⋅ denotes the standard inner product in \(\mathbb{R}^{N}\). Define the subsets of Ω dependent on \(p_{i}\) by \(U_{p_{i}}:= \{ x\in \Omega : p(x)\geq 2 \} \) and \(V_{p_{i}}:= \{ x\in \Omega : 1< p(x)< 2 \} \). For \(i=1\), respectively replacing ϱ and ζ by \(\nabla u_{1_{m}}\) and \(\nabla u_{1_{k}}\) when \(j=1\), and by \(u_{1_{m}}\) and \(u_{1_{k}}\) when \(j=2\), in the first line of relation (3.23), and integrating over Ω, we obtain
On the other hand, by the second line of relation (3.23), we have
where \(\sigma _{1}(x) =(|\nabla u_{1_{m}}|+|\nabla u_{1_{k}}|)\) and \(\sigma _{2}(x) =(|u_{1_{m}}|+| u_{1_{k}}|)\). Hence, by Hölder’s inequality and Lemma 2.2,
Since \(\lbrace u_{1_{m}} \rbrace \) is a bounded sequence in \(W^{1,h_{1}(x)}(\Omega )\cap W^{1,p_{1}(x)} \), we have
hence \(\lbrace u_{1_{m}} \rbrace \) is a Cauchy sequence in \(W^{1,p_{1}(x)}\cap W^{1,h_{1}(x)}(\Omega )\). We argue similarly for \(\lbrace u_{i_{m}} \rbrace \),
Thus, we can conclude that \(u_{m}= (u_{1_{m}},\dots ,u_{n_{m}})\to u= (u_{1},\dots ,u_{n})\) strongly in X as \(m\to +\infty \). Therefore, we have that \(E_{\theta , \lambda}(u)= c_{\theta , \lambda}>0\) and \(E_{\theta , \lambda}'(u)=0\) in \(X'\), i.e., \(u\in X\) is a weak solution of problem (3.1). Since \(E_{\theta , \lambda}(u)=c_{\theta , \lambda}>0=E_{\theta , \lambda}(0)\), we can conclude that \(u\not \equiv 0\). □
4 Proof of the main theorem
Now we are in position to prove Theorem 1.2.
Proof
Invoking Theorem 3.1, for all \(\lambda \geq \lambda _{*}\) let \(u_{\lambda}=(u_{1,\lambda},u_{2,\lambda},\dots ,u_{n,\lambda})\) be a solution of system (3.1). We shall prove that
where \(\tau _{i}^{0}\) is defined as at the beginning of Sect. 3. We argue by contradiction and suppose that there is a sequence \(\{\lambda _{m} \}_{m\in \mathbb{N}}\subset \mathbb{R}\) such that \(\mathcal{A}_{i}(u_{i,\lambda _{m}})\geq \tau _{i}^{0}\), for all \(i \in \{1,2,\dots ,n\}\). By assumption \((\boldsymbol{A}_{1})\) and the fact \(\mathcal{A}_{i}(u_{i,\lambda _{m}})\geq \tau _{i}^{0}\), we get
Since \(u_{\lambda _{m}}=(u_{1,\lambda _{m}},\dots ,u_{n,\lambda _{m}})\) is a critical point of the functional \(E_{\theta , \lambda _{m}}\), we can conclude, using assumptions \((\boldsymbol{M})\) and \((\boldsymbol{F}_{3})\), that
If \(k_{i}^{3}=0\), then, using relation (4.2), we find \(\int _{\Omega} (|\nabla u_{i}| ^{p_{i}(x)} + |u_{i}| ^{p_{i}(x)} ) \,dx \geq \frac{\tau _{i}^{0}}{\max \{k_{1_{i}}^{1},k_{2_{i}}^{1}\}}\), thus we have
This contradicts Lemma 3.4, because \(\lim_{m\to +\infty }c_{\theta ,\lambda _{m}}=0\). On the other hand, if \(k_{i}^{3}>0\), we multiplying relation (4.3) by \(\max_{1\leq i\leq n}\{\max \{k_{1_{i}}^{1},k_{2_{i}}^{1} \}\times k_{i}^{3}\}>0\), and, by using also relation (4.2), we get
where \(\kappa _{i}=\min \{\max \{k_{1_{i}}^{0},k_{2_{i}}^{0}\}\times k_{i}^{3}, \max \{k_{1_{i}}^{1},k_{2_{i}}^{1}\}\times \max \{k_{1_{i}}^{2},k_{2_{i}}^{2} \} \}\). This also contradicts Lemma 3.4 because \(\lim_{m\to +\infty }c_{\theta ,\lambda}=0\). Hence, we can conclude in both cases that there exists \(\lambda ^{\ast}\geq \lambda _{\ast}\) such that \(\mathcal{A}_{i}(u_{i,\lambda})\geq \tau _{i}^{0}\), for all \(\lambda \geq \lambda ^{\ast}\). So, we can find \(M_{\theta _{i}}(\mathcal{A}_{i}(u_{\lambda}))=M_{\theta _{i}}( \mathcal{A}_{i}(u_{\lambda}))\), for all \(\lambda \geq \lambda ^{\ast}\), which implies that \(E_{\theta ,\lambda}(u_{\lambda})=E_{\lambda}(u_{\lambda})\) and \(E_{\theta ,\lambda}'(u_{\lambda})=E_{\lambda}'(u_{\lambda})\), that is, \(u_{\lambda}\) is a nontrivial weak solution of the problem (1.1), for each \(\lambda \geq \lambda ^{\ast}\).
It now remains to consider the asymptotic behavior of solutions to problem (1.1). By assumptions \((\boldsymbol{A}_{2})\), \((\boldsymbol{A}_{4})\), \((\boldsymbol{M})\), \((\boldsymbol{F}_{3})\), and inequalities (2.2)–(2.3) and (3.2), arguing as above, we obtain
Hence, by Lemma 3.4, we get \(\lim_{\lambda \to +\infty } \|u_{\lambda}\|= \lim_{ \lambda \to + \infty }\max_{1\leq i\leq n} \{\| u_{i}\|_{1,p_{i}(x)}+ \mathcal{K}(k_{i}^{3}) \| u_{i}\|_{1,q_{i}(x)} \}= 0\). □
5 Some examples
In the last section, we shall exhibit some examples which are interesting from the mathematical point of view and have a wide range of applications in physics and other scientific fields that fall within the general class of systems studied in this paper, under adequate assumptions on functions \(a_{i_{j}}\).
Example 5.1
Taking \(a_{1_{i}}\equiv 1\) and \(a_{2_{i}}\equiv 1\), we see that \(a_{1_{i}}\) satisfies the assumptions \((\boldsymbol{A}_{1})\), \((\boldsymbol{A}_{2})\), and \((\boldsymbol{A}_{3})\), with \(k^{0}_{j_{i}}=k^{1}_{j_{i}}=1\), \(k^{2}_{j_{i}}>0\), and \(k^{3}_{i}=0\), for all \(i \in \{1,2,\dots ,n\}\) and \(j=1\) or 2. Hence, system (1.1) becomes
for \(1\leq i\leq n\) (\(n\in \mathbb{N}^{\ast}\)), where
The operator \(\Delta _{p_{i}(x)}u_{i}:=\operatorname{div}(|\nabla u_{i}|^{p_{i}(x)-2} \nabla u_{i})\) is the so-called \(p_{i}(x)\)-Laplacian, which coincides with the usual \(p_{i}\)-Laplacian when \(p_{i}(x)=p_{i}\), and with the Laplacian when \(p_{i}(x)=2\).
Example 5.2
Taking \(a_{j_{i}}(\xi )= 1+ \xi ^{\frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}}\), we see that \(a_{j_{i}}\) satisfies the assumptions \((\boldsymbol{A}_{1})\), \((\boldsymbol{A}_{2})\), and \((\boldsymbol{A}_{3})\), with \(k^{0}_{j_{i}}=k^{1}_{j_{i}}=k^{2}_{j_{i}}=k^{3}_{i}=1\), for all \(i\in \{1,2,\dots ,n\}\) and \(j=1\) or 2. Hence, system (1.1) becomes the following \((p,q)\)-Laplacian system
for \(1\leq i\leq n\) (\(n\in \mathbb{N}^{\ast}\)), where
As explained in Cherfils and Il’yasov [16], the study of system (5.2) was motivated by the following more general reaction–diffusion system:
which has applications in biophysics (see, e.g., Fife [25], Murray [40]), plasma physics (see, e.g., Wilhelmsson [49]), and chemical reactions design (see, e.g., Aris [4]). In these applications, u represents concentration, \(\operatorname{div}[H(u)\nabla u ]\) is the diffusion with a diffusion coefficient, and the reaction term \(d(x,u)\) relates to the source and loss processes. For further details, we refer the interested reader to, e.g., Mahshid and Razani [38], He and Li [30], and the references therein.
We continue with other examples which are also interesting from the mathematical point of view.
Example 5.3
Taking \(a_{j_{i}}(\xi )= 1+ \frac{\xi}{\sqrt{1+\xi ^{2}}}\) and \(a_{2_{i}}\equiv 1\), we see that \(a_{j_{i}}\) satisfies the assumptions \((\boldsymbol{A}_{1})\), \((\boldsymbol{A}_{2})\), and \((\boldsymbol{A}_{3})\), with \(k^{0}_{1_{i}}=k^{0}_{2_{i}}=k^{1}_{2_{i}}=1\), \(k^{1}_{1_{i}}=2\), and \(k^{3}_{i}=0\), \(k^{2}_{1_{i}}>0\), and \(k^{2}_{2_{i}}>0\), for all \(i\in \{1,2,\dots ,n\}\). Hence, system (1.1) becomes
for \(1\leq i\leq n\) (\(n\in \mathbb{N}^{\ast}\)), where
The operator \(\operatorname{div} ( (1+ \frac{|\nabla u|^{p(x)}}{\sqrt{1+|\nabla u|^{2p(x)}}} )|\nabla u|^{p(x)-2} \nabla u )\) is said to be \(p_{i}(x)\)-Laplacian-like or is called a generalized capillary operator. The capillarity can be briefly explained by considering the effects of two opposing forces: adhesion, i.e., the attractive (or repulsive) force between the molecules of the liquid and those of the container; and cohesion, i.e., the attractive force between the molecules of the liquid. The study of the capillary phenomenon has gained much attention. This increasing interest is motivated not only by the fascination in naturally occurring phenomena, such as motion of drops, bubbles, and waves, but also by its importance in applied fields, raging from industrial and biomedical and pharmaceutical to microfluidic systems; for further details, we refer the interested reader to, e.g., Ni and Serrin [41], and the references therein.
Example 5.4
Taking \(a_{1_{i}}(\xi )= 1+ \frac{1}{(1+\xi )^{\frac{p_{i}(x)-2}{p_{i}(x)}}}\) and \(a_{2_{i}}\equiv 1\), we see that \(a_{j_{i}}\) satisfies the assumptions \((\boldsymbol{A}_{1})\), \((\boldsymbol{A}_{2})\), and \((\boldsymbol{A}_{3})\), with \(k^{0}_{1_{i}}=k^{0}_{2_{i}}=k^{1}_{2_{i}}=1\), \(k^{1}_{1_{i}}=2\), \(k^{3}_{i}=0\), \(k^{2}_{1_{i}}>0\), and \(k^{2}_{2_{i}}>0\), for all \(i\in \{1,2,\dots ,n\}\). Hence, system (1.1) becomes
for \(1\leq i\leq n\) (\(n\in \mathbb{N}^{\ast}\)), where
Example 5.5
Taking \(a_{1_{i}}(\xi )= 1+ \xi ^{ \frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}} + \frac{1}{(1+\xi )^{\frac{p_{i}(x)-2}{p_{i}(x)}}}\) and \(a_{2_{i}}(\xi )= 1+ \xi ^{ \frac{q_{i}(x)-p_{i}(x)}{p_{i}(x)}}\), we see that \(a_{1_{i}}\) satisfies the assumptions \((\boldsymbol{A}_{1})\), \((\boldsymbol{A}_{2})\), \((\boldsymbol{A}_{3})\), and \((\boldsymbol{H}_{3})\), with \(k^{0}_{1_{i}}=k^{0}_{2_{i}}=k^{1}_{2_{i}}=1\), \(k^{1}_{1_{i}}= 2\), and \(k^{3}_{i}=k^{2}_{1_{i}}=k^{2}_{2_{i}}=1\), for all \(i\in \{1,2,\dots ,n\}\). Hence, system (1.1) becomes
for \(1\leq i\leq n\) (\(n\in \mathbb{N}^{\ast}\)), where
On the other hand, the class of systems (1.1) can contain one model of the above divergence operators, as in Examples 5.1–5.5, or many different models of divergence operators simultaneously, depending on the phenomenon studied. Moreover, each equation in this class can also be degenerate or nondegenerate.
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Chems Eddine, N., Repovš, D.D. The Neumann problem for a class of generalized Kirchhoff-type potential systems. Bound Value Probl 2023, 19 (2023). https://doi.org/10.1186/s13661-023-01705-6
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DOI: https://doi.org/10.1186/s13661-023-01705-6