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Nonlinear boundary value problems of a class of elliptic equations involving critical variable exponents
Boundary Value Problems volume 2019, Article number: 122 (2019)
Abstract
In this paper, we first obtain the existence of solutions for a class of elliptic equations involving critical variable exponents and nonlinear boundary values by the mountain pass theorem and concentration compactness principle. Then, under suitable assumptions, we obtain a sequence of solutions with positive energies going towards infinity by Fountain Theorem.
1 Introduction
In the studies of electrorheological fluids, nonlinear elasticity, and image restoration in practical applications, the classical Lebesgue and Sobolev spaces are inapplicable; see [1,2,3]. Such problems are inhomogeneous and nonlinear with variable exponential growth conditions. So we need to study the problems based on the theory of variable exponent Lebesgue and Sobolev spaces.
Since Kováčik and Rákosník first studied the \(L^{p(x)}\) spaces and \(W^{k,p(x)}\) spaces in [4], a lot of research has been done concerning these kinds of variable exponent spaces. The existence of solutions for \(p(x)\)-Laplacian Dirichlet problems on bounded domains has been widely discussed. For example, in [5] and [6], some results as regards the existence of solutions under some conditions are obtained.
The nonlinear elliptic boundary value problems appear when we study the conformal deformations on Riemannian manifolds with boundary. The study of nonlinear elliptic boundary value problems with p-Laplacian has become an interesting topic in recent years. Many results have been obtained on this kind of problems; see [7,8,9]. In the fractional Laplacian setting, the existence of solutions for the problem has been obtained; see [10,11,12,13,14].
But at present there are few papers on the study of nonlinear elliptic boundary value problems with \(p(x)\)-Laplacian. So this topic is worth further discussing.
In this paper, we consider the problem
where \(\varOmega \subset \mathbb{R}^{N}\) is a bounded domain with smooth boundary, \(p(x)\) is Lipschitz continuous and satisfies \(1< p_{1} \leq p(x)\leq p_{2}<N\), \(p^{*}(x)=\frac{Np(x)}{N-p(x)}\). We assume that \(a:\overline{\varOmega }\times \mathbb{R}^{N}\rightarrow \mathbb{R}\) is a Carathéodory function and we have the continuous derivative with respect to η of a function \(A:\overline{\varOmega }\times \mathbb{R}^{N}\rightarrow \mathbb{R}\). Suppose that a and A satisfy the following hypotheses:
- (A1):
-
For \(\forall x\in \overline{\varOmega }\), the equality \(A(x,0)=0\) holds.
- (A2):
-
There exists a positive constant \(c_{0}\) such that
$$ \bigl\vert a(x,\eta ) \bigr\vert \leq c_{0} \bigl(1+ \vert \eta \vert ^{p(x)-1} \bigr) $$for all \(x\in \overline{\varOmega }\) and \(\eta \in \mathbb{R}^{N}\).
- (A3):
-
For all \(x\in \overline{\varOmega }\) and \(\eta _{1},\eta _{2}\in \mathbb{R}^{N}\), the following inequality holds:
$$ 0\leq \bigl[a(x,\eta _{1})-a(x,\eta _{2}) \bigr]\cdot ( \eta _{1}-\eta _{2}), $$where equality holds if and only if \(\eta _{1}=\eta _{2}\).
- (A4):
-
For all \(x\in \overline{\varOmega }\) and \(\eta \in \mathbb{R}^{N}\), the inequalities
$$ \vert \eta \vert ^{p(x)}\leq a(x,\eta )\cdot \eta \leq p(x)A(x,\eta ) $$hold true.
- (A5):
-
For all \(x\in \overline{\varOmega }\) and \(\eta \in \mathbb{R}^{N}\), the equality \(A(x,-\eta )=A(x,\eta )\) holds true.
The above type of assumptions can be found in other papers too; for example, see [15, 16]. But in [16], the authors establish the existence of a solution for an elliptic problem with Dirichlet boundary conditions, and in [15], the authors consider the subcritical case. In the present paper, the problem involves not only the critical Sobolev exponents, but also the nonlinear boundary conditions. Because of the critical exponents, the compactness of the embedding fails, so to recover the loss of the compactness, we use the concentration compactness principle in [17].
Throughout this paper, we assume that the following conditions hold:
- (F1):
-
\(f\in C(\overline{\varOmega }\times \mathbb{R})\), \(f(x,0)\equiv 0\) and \(\vert f(x,t)\vert \leq C_{1}(1+\vert t\vert ^{\alpha _{1}(x)-1})\), \(\alpha _{1}\in C(\overline{ \varOmega })\) with \(p(x)\ll \alpha _{1}(x)\ll p^{*}(x)\), and \(F(x,t)>0\) in \(\varOmega _{0}\times \mathbb{R}\) for some nonempty open set \(\varOmega _{0} \subset \varOmega \), where \(C_{1}\) is a positive constant.
- (F̃1):
-
\(f\in C(\overline{\varOmega }\times \mathbb{R})\), \(\vert f(x,t)\vert \leq C_{1}(1+\vert t\vert ^{\alpha _{1}(x)-1})\), \(\alpha _{1}\in C(\overline{\varOmega })\) with \(1\leq \alpha _{1}(x)\ll p(x)\) and \(F(x,t)>0\) in \(\varOmega _{0}\times \mathbb{R}\) for some nonempty open set \(\varOmega _{0}\subset \varOmega \).
- (F2):
-
\(f(x,t)=-f(x,-t)\) for any \((x,t)\in \overline{\varOmega }\times \mathbb{R}\).
- (F3):
-
For any \((x,t)\in \overline{\varOmega }\times \mathbb{R}\), there exists a function \(\mu _{1}(x)\in C^{1}(x)\) such that \(\mu _{1}(x) \gg p(x)\) and \(0\leq \mu _{1}(x)F(x,t)\leq f(x,t)t\), where \(F(x,t)=\int ^{t}_{0}f(x,s)\,ds\).
- (F4):
-
\(f(x,t)=o(\vert t\vert ^{p(x)-1})\) hold uniformly for any \(x\in \overline{ \varOmega }\), as \(t\rightarrow 0\).
- (B1):
-
\(b\in C(\overline{\varOmega }\times \mathbb{R})\), \(b(x,0)\equiv 0\) and \(\vert b(x,t)\vert \leq C_{2}\vert t\vert ^{\alpha _{2}(x)-1}\), \(\alpha _{2}\in C(\overline{ \varOmega })\) with \(p(x)\ll \alpha _{2}(x)\ll p_{*}(x)\), and \(B(x,t)>0\) in \(\partial \varOmega \times \mathbb{R}\), where \(C_{2}\) is a positive constant and \(p_{*}(x)=\frac{(N-1)p(x)}{N-p(x)}\).
- (B̃1):
-
\(b\in C(\overline{\varOmega }\times \mathbb{R})\), \(\vert b(x,t)\vert \leq C_{2}(1+\vert t\vert ^{\alpha _{2}(x)-1})\), \(\alpha _{2}\in C(\overline{\varOmega })\) with \(p(x)\ll \alpha _{2}(x) \ll p_{*}(x)\) and \(B(x,t)>0\) in \(\partial \varOmega \times \mathbb{R}\).
- (B2):
-
\(b(x,t)=-b(x,-t)\) for any \((x,t)\in \overline{\varOmega }\times \mathbb{R}\).
- (B3):
-
For any \((x,t)\in \partial \varOmega \times \mathbb{R}\), there exists a function \(\mu _{2}(x)\in C^{1}(x)\) such that \(\mu _{2}(x)\gg p(x)\) and \(0\leq \mu _{2}(x)B(x,t)\leq b(x,t)t\), where \(B(x,t)=\int ^{t}_{0}b(x,s)\,ds\).
- (H1):
-
For any \(x\in \varOmega \), there exists \(h_{1}>0\) such that \(h(x)\geq h_{1}\) and \(h(x)\in L^{\infty }(\varOmega )\).
2 Preliminaries
We first recall some facts on spaces \(L^{p(x)}\) and \(W^{k,p(x)}\). For details see [4, 18, 19].
Let \(\mathbf{P}(\varOmega )\) be the set of all Lebesgue measurable functions \(p:\varOmega \to [1,\infty ]\), we denote
where \(\varOmega _{\infty }=\{x\in \varOmega :p(x)=\infty \}\).
The variable exponent Lebesgue space \(L^{p(x)}(\varOmega )\) is the class of all functions u such that \(\rho _{p(x)}(tu)<\infty \), for some \(t>0\). \(L^{p(x)}(\varOmega )\) is a Banach space equipped with the norm
For any \(p\in \mathbf{P}(\varOmega )\), we define the conjugate function \(p'(x)\) as
Theorem 2.1
Let \(p\in \mathbf{P}(\varOmega )\). For any \(u\in L^{p(x)}(\varOmega )\) and \(v\in L^{p'(x)}(\varOmega )\),
For any \(p\in \mathbf{P}(\varOmega )\), we denote
and we denote by \(p(x)\ll q(x)\) the fact that \(\inf_{x\in \varOmega } (q(x)-p(x))>0\).
Theorem 2.2
Let \(p\in \mathbf{P}(\varOmega )\) with \(p_{2}<\infty \). For any \(u\in L^{p(x)}(\varOmega )\), we have
-
(1)
if \(\Vert u\Vert _{L^{p(x)}}\geq 1\), then \(\Vert u\Vert _{L^{p(x)}}^{p_{1}} \leq \int _{\varOmega } \vert u\vert ^{p(x)} \,dx\leq \Vert u\Vert _{L^{p(x)}}^{p_{2}}\);
-
(2)
if \(\Vert u\Vert _{L^{p(x)}}<1\), then \(\Vert u\Vert _{L^{p(x)}}^{p_{2}} \leq \int _{\varOmega } \vert u\vert ^{p(x)} \,dx\leq \Vert u\Vert _{L^{p(x)}}^{p_{1}}\).
The variable exponent Sobolev space \(W^{1,p(x)}(\varOmega )\) is the class of all functions \(u\in L^{p(x)}(\varOmega )\) such that \(\vert \nabla u\vert \in L ^{p(x)}(\varOmega )\). \(W^{1,p(x)}(\varOmega )\) is a Banach space equipped with the norm
For \(u\in W^{1,p(x)}(\varOmega )\), if we define
then \(|\!|\!|\cdot |\!|\!|\) and \(\Vert \cdot \Vert _{W^{1,p(x)}}\) are equivalent norms on \(W^{1,p(x)}(\varOmega )\). In fact, we have
Theorem 2.3
For any \(u\in W^{1,p(x)}(\varOmega )\), we have
-
(1)
if \(|\!|\!|u|\!|\!|\geq 1\), then \(|\!|\!|u|\!|\!|^{p_{1}}\leq \int _{\varOmega } (\vert \nabla u\vert ^{p(x)}+\vert u\vert ^{p(x)}) \,dx\leq |\!|\!|u|\!|\!|^{p_{2}}\);
-
(2)
if \(|\!|\!|u|\!|\!|<1\), then \(|\!|\!|u|\!|\!|^{p_{2}}\leq \int _{\varOmega } (\vert \nabla u\vert ^{p(x)}+\vert u\vert ^{p(x)}) \,dx\leq |\!|\!|u|\!|\!|^{p_{1}}\).
Theorem 2.4
Let Ω be a bounded domain with the cone property. If \(p\in C(\bar{\varOmega })\) satisfying \(1< p_{1}\leq p(x)\leq p_{2}<N\) and q is a measurable function defined on Ω with
then the embedding \(W^{1,p(x)}(\varOmega )\hookrightarrow L^{q(x)}( \varOmega )\) is compact.
Theorem 2.5
Let Ω be a domain with the cone property. If p is Lipschitz continuous and satisfies \(1< p_{1}\leq p(x)\leq p_{2}<N\), q is a measurable function defined on Ω with
then the embedding \(W^{1,p(x)}(\varOmega )\hookrightarrow L^{q(x)}( \varOmega )\) is continuous.
Theorem 2.6
Let \(\varOmega \subset \mathbb{R}^{N}\) be an open bounded domain with Lipschitz boundary. Suppose that \(p\in C(\bar{\varOmega })\) and \(1< p_{1}\leq p(x)\leq p_{2}<N\). If \(q\in C(\partial \varOmega )\) satisfies the condition
then the boundary trace embedding \(W^{1,p(x)}(\varOmega )\rightarrow L ^{q(x)}(\partial \varOmega )\) is compact.
In the proof of the main results, we will use the following principle of concentration compactness in \(W^{1,p(x)}(\varOmega )\), established in [17].
Theorem 2.7
Assume that p is Lipschitz continuous on Ω̄ and satisfies \(1< p_{1}\leq p(x)\leq p_{2}<N\), and Ω is a bounded domain in \(\mathbb{R}^{N}\). Let \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) with \(\Vert \nabla u_{n}\Vert _{L^{p(x)}}\leq 1\) such that
as \(n\to \infty \). Denote
Then the limit measures are of the form
where J is a countable set, \(\{\mu _{j}\},\{\nu _{j}\}\subset [0, \infty )\), \(\{x_{j}\}\subset \bar{\varOmega }\), \(\widetilde{\mu }\in M( \varOmega )\) is a non-atomic nonnegative measure. The atoms and the regular part satisfy the generalized Sobolev inequality
where \(p^{*}_{1}=\inf_{x\in \varOmega } p^{*}(x)\), \(p^{*}_{2}=\sup_{x \in \varOmega } p^{*}(x)\).
3 Existence of solutions for the problems
Set
We say that \(u\in W^{1,p(x)}(\varOmega )\) is a weak solution of \(p(x)\)-Laplacian problem (1.1), if, for any \(v\in W^{1,p(x)}( \varOmega )\),
So next we need only to consider the existence of nontrivial critical points of \(I(u)\).
Lemma 3.1
([16], Lemma 1)
The functional Λ is well-defined on \(W^{1,p(x)}(\varOmega )\), and for all \(u,v\in W^{1,p(x)}\),
Lemma 3.2
([5], Lemma 2.9)
Suppose that f satisfies (F1) or (F̃1). Then \(K(u)\) is weakly continuous.
Lemma 3.3
([5], Theorem 2.10)
Suppose that f satisfies (F1) or (F̃1). Then \(K(u)\) is differentiable on \(W^{1,p(x)}\), and, for all \(u,v\in W^{1,p(x)}\),
In the same way, the function L leads to a conclusion similar to Lemma 3.2 and Lemma 3.3.
Lemma 3.4
([20], Theorem 4.1)
The mapping a is an operator of type \({S_{+}}\), that is, if \(u_{n}\to u\) weakly in \(W^{1,p(x)}(\varOmega )\) and
then \(u_{n}\to u\) strongly in \(W^{1,p(x)}(\varOmega )\).
Theorem 3.1
Assume hypotheses (F1), (F3), (F4), (B1), (B3) and (H1) are fulfilled. Then there exists \(M>0\) such that, whenever \(h(x)\leq M\), the problem has a nontrivial solution.
Proof
(1) There exists \(r>0\) such that \(\inf \{I(u):|\!|\!|u|\!|\!|=r, u\in W ^{1,p(x)}(\varOmega )\}>c\).
From (F1), (F4) and (B1) we have
Next, from (A4),
Let \(\varepsilon <\frac{1}{2p_{2}}\), we get
As \(\alpha _{2}(x)\), \(p(x)\) are continuous on Ω̅, there exists \(\delta _{1}>0\) such that \(\vert \alpha _{2}(x)-\alpha _{2}(y)\vert <\varepsilon \) and \(\vert p(x)-p(y)\vert <\varepsilon \) for any \(\varepsilon \in (0,1)\) whenever \(\vert x-y\vert <\delta _{1}\). Take \(x\in \overline{\varOmega }\), for any \(y\in B_{\delta _{1}(x)}(x)\cap \overline{\varOmega }\), we have
and
As \(p(x)\ll \alpha _{2}(x)\), take \(\varepsilon =\frac{1}{4}\inf_{x \in \overline{\varOmega }}(\alpha _{2}(x)-p(x))\), we have
then
and further
In the same manner, we get
\(\{B_{\delta _{x}}(x),x\in \overline{\varOmega }\}\) is an open covering of Ω̅. Since Ω̅ is compact, we can pick a finite subcovering \(\{B_{\delta _{i}}(x_{i})\}_{i=1}^{k}\) for Ω̅ from the covering \(\{B_{\delta _{x}}(x),x\in \overline{ \varOmega }\}\) such that \(\bigcup_{i=1}^{k} B_{\delta _{i}}(x_{i})\supset \overline{\varOmega }\). Denote \(\delta _{l}=\min \{ \delta _{i},i=1,2,\ldots,k\}\), we can use all the hypercubes whose length of the side is \(\frac{\delta _{l}}{2}\) to divide the entire space \(\mathbb{R}^{N}\), then \(\bigcup_{i=1}^{k} B_{\delta _{i}}(x_{i})\cap \varOmega \) is divided by finite open regions \(\{\varOmega _{i}\}_{i=1}^{m}\) which mutually have no common points, and \(\overline{\varOmega }=\bigcup_{i=1}^{m}\overline{ \varOmega _{i}}\). Then
By Theorems 2.5 and 2.6, we know that there exist \(c_{4},c_{5}>1\) such that
where \(i=1,2,\ldots,m\).
Take \(|\!|\!|u|\!|\!|\leq [\max (c_{4},c_{5})]^{-1}\), then \(|\!|\!|u|\!|\!|_{\varOmega _{i}}<[\max (c_{4},c_{5})]^{-1}\) and
then we have
Let
By (3.3), there exists \(0< t_{i}<1\) such that \(g(t)\) is positive and increasing for any \(t\in (0,t_{i}]\).
Take \(t_{k}=\min \{t_{i},i=1,2,\ldots,m\}\). Since \(|\!|\!|u|\!|\!|\leq \sum_{i=1} ^{m} |\!|\!|u|\!|\!|_{\varOmega _{i}}\), when \(|\!|\!|u|\!|\!|=r< t_{k}\), there exists j such that \(\frac{r}{m}\leq |\!|\!|u|\!|\!|_{\varOmega _{j}}\leq r< t _{j}\), then
Take
we have \(I(u)\geq c\), where \(c=\frac{1}{4p_{2}}(\frac{r}{m})^{p_{j} ^{+}}\).
(2) There exists \(e\in W^{1,p(x)}(\varOmega )\) such that \(|\!|\!|e|\!|\!|>r\), then we have \(I(e)<0\).
From (F1) and (F3), we have
for any \((x,t)\in {\varOmega _{0}}\times \mathbb{R}\).
Next from (A1) and (A2), for any \(x\in \overline{\varOmega }\),
and
Pick \(x_{0}\in \varOmega _{0}\). As \(\mu _{1}\), p is continuous on Ω̅, there exists \(0<2R<1\) such that
for \(B_{2R}(x_{0})\subset \varOmega _{0}\). Let \(\phi \in C_{0}^{\infty }(B _{2R}(x_{0}))\) such that \(\phi \equiv 1\) for any \(x\in B_{2R}(x_{0})\), \(0 \leq \phi \leq 1\) and \(\vert \nabla \phi \vert \leq \frac{1}{R}\). Then, for \(s>1\),
where \(\overline{C}=\frac{C\int _{B_{2R}(x_{0})}\vert \phi \vert ^{\mu _{1}(x)}\,dx}{\vert B _{2R}(x_{0})\vert }\).
As \(\phi \equiv 1\) for any \(x\in B_{2R}(x_{0})\), \(\int _{B_{2R}(x_{0})}\vert \phi \vert ^{\mu _{1}(x)}\,dx>0\), thus \(\overline{C}>0\).
As \(p(x)>1\), if s is sufficiently large, then \(s^{1-p(x)}<1\). Thus
Because \(\mu _{1x_{0}}^{-}-p_{2x_{0}}>0\), when s is sufficiently large, we have \(|\!|\!|s\phi |\!|\!|>r\) and \(I(s\phi )<0\).
(3) The functional I satisfies the (PS) condition (i.e. any sequence \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) with \(I(u_{n})\leq c\) and \(I^{\prime }(u_{n})\rightarrow 0\) as \(i\rightarrow \infty \) in \(W^{-1,p^{\prime }(x)}\) possesses a convergent subsequence).
(i) First, we show that the (PS) sequence \(\{u_{n}\}\subset W^{1,p(x)}\) is bounded.
Note that \(p(x)\) is Lipschitz continuous, then there exists a Lipschitz continuous function \(v(x)\) such that \(p(x)\ll v(x)\leq p^{*}(x)\) and
Take
we obtain
where \(l_{1}=\inf_{x\in \varOmega }\{v(x)-p(x)\}\), \(l_{2}=\inf_{x \in \varOmega }{p^{*}(x)-v(x)\vert }\), \(M= \sup_{x\in \varOmega }\vert \nabla v(x)\vert \).
By the Young inequality, we have
Take \(\varepsilon _{1}=\min \{1,\frac{v_{1}^{2}p_{1}l_{1}}{2c_{0}Mv _{2}p_{2}}\}\) such that \(\frac{c_{0}M\varepsilon _{1}}{v_{1}^{2}p_{1}} \leq \frac{l_{1}}{2v_{2}p_{2}}\).
By the Young inequality, we have
Take \(\varepsilon _{2}=\min \{1,\frac{v_{1}^{2}l_{1}}{2c_{0}Mv_{2}p _{2}}\}\) such that \(\frac{c_{0}M\varepsilon _{2}}{v_{1}^{2}}\leq \frac{l _{1}}{2v_{2}p_{2}}\).
By the Young inequality again, we have
Take \(\varepsilon _{3}=\min \{1,\frac{h_{1}l_{2}v_{1}^{2}p_{1}p_{1} ^{*}}{2c_{0}Mv_{2}p_{2}^{*}p_{2}\varepsilon _{2}^{1-p_{2}}}\}\) such that \(\frac{c_{0}Mp_{2}\varepsilon _{2}^{1-p_{2}}}{v_{1}^{2}p_{1}p_{1}^{*}} \leq \frac{l_{1}}{2v_{2}p_{2}}\). Then
As
we have \(\Vert \frac{u_{n}}{v}\Vert _{L^{p(x)}}\leq \frac{ \Vert u_{n}\Vert _{L^{p(x)}}}{v_{1}}\). Since
we have
so
where C is constant. Moreover, \(\frac{\Vert u_{n} \Vert _{W^{1,p(x)}}}{2}\leq |\!|\!|u|\!|\!|\leq 2\Vert u_{n} \Vert _{W^{1,p(x)}}\), we have
when n is sufficiently large, we obtain
By the Young inequality, we have
Take \(\varepsilon =\frac{l_{1}p_{1}}{4v_{2}p_{2}C}\) such that
then \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) is bounded.
(ii) Next, we show that the (PS) sequence \(\{u_{n}\}\subset W^{1,p(x)}( \varOmega )\) possesses a convergent subsequence.
We know that \(\{u_{n}\}\) is bounded. As \(W^{1,p(x)}(\varOmega )\) is reflexive, passing to a subsequence (still denoted by \(\{u_{n}\}\)), we may assume that there exists \(u\in W^{1,p(x)}(\varOmega )\) such that \(u_{n}\rightarrow u\) weakly in \(W^{1,p(x)}\) and \(u_{n}\rightarrow u\) a.e. on Ω.
From the definition of (PS) sequence, we obtain \(\lim_{n\rightarrow \infty }\langle I'(u_{n}),u_{n}-u\rangle =0\), i.e.
As \(p(x)< p^{*}(x)\), the embedding \(W^{1,p(x)}\rightarrow L^{p(x)}( \varOmega )\) is compact, so \(u_{n}\rightarrow u\) strongly in \(L^{p(x)}( \varOmega )\). Hence when \(n\rightarrow \infty \),
If we could verify that \(u_{n}\rightarrow u\) strongly in \(L^{p^{*}(x)}( \varOmega )\), we can obtain
Therefore, \(\lim_{n\rightarrow \infty }\int _{\varOmega }a(x, \nabla u_{n})\,dx=0\), by Lemma 3.4, a is a \({S}_{+}\) type operator, then \(u_{n}\rightarrow u\) strongly in \(L^{p^{*}(x)}( \varOmega )\). □
Next, in order to complete Theorem 3.1, we prove the following lemma.
Lemma 3.5
Let the assumptions of Theorem 3.1 be satisfied. If the (PS) sequence \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) is bounded, then there exists \(M>0\) such that whenever \(h(x)\leq M\), \(u_{n}\rightarrow u\) strongly in \(L^{p^{*}(x)}(\varOmega )\).
Proof
As \(u_{n}\rightarrow u\) strongly in \(L^{p(x)}(\varOmega )\), there exists subsequence (still denoted by \(\{u_{n}\}\)), \(u_{n}\rightarrow u\) a.e. on Ω. Note that \(\{u_{n}\}\subset W^{1,p(x)}(\varOmega )\) is bounded, by Borel measure theory, we may assume that
\(M(\overline{\varOmega })\) is the space of finite nonnegative Borel measures on Ω.
From the principle of concentration compactness,
where J is a countable set, \(\{x_{j},j\in J\}\subset \overline{ \varOmega }\), \(\{\nu _{j}\}\subset [0,+\infty )\), \(\delta _{x_{j}}\) is a measure concentrating upon \(x_{j}\), μ̃ is a nonnegative non-atomic measure.
a. First, we show that \(\mu (\{x_{j}\})=\nu (\{x_{j}\})=0\) for any \(j\in J\).
As Ω̅ is compact, so we only need to verify, for any \(x\in \overline{\varOmega }\), there exists \(r_{0}>0\) such that \(\mu (\{x_{j}\})=\nu (\{x_{j}\})=0\) for \(x_{j}\in \overline{\varOmega } \cap B_{r_{0}}(x)\).
Note that \(p(x)\) is Lipschitz continuous and \(p(x)\ll p^{*}(x)\), there exists \(r_{0}>0\) such that
For any \(\varepsilon >0\), let \(\phi _{\varepsilon }\in C_{0}^{\infty }(B _{2\varepsilon }(x_{j}))\) such that \(\phi \equiv 1\) for any \(x\in B_{2\varepsilon }(x_{j})\), \(0\leq \phi _{\varepsilon }\leq 1\) and \(\vert \nabla \phi _{\varepsilon }\vert \leq \frac{2}{\varepsilon }\). Note that
Since \(u_{n}\in W^{1,p(x)}(\varOmega )\), \(\{u_{n}\phi _{\varepsilon }\}\) is bounded on \(W^{1,p(x)}(\varOmega )\), we have \(\langle I'(u_{n}),u_{n} \phi _{\varepsilon }\rangle \rightarrow 0\) as \(n\rightarrow \infty \). Note that
Since \(\vert f(x,u_{n}) \vert \leqslant C_{1}(1+ \vert u_{n} \vert ^{\alpha _{1}(x)-1})\), \(\vert f(x,u_{n})u_{n} \vert \leqslant C _{1}(1+ \vert u_{n} \vert ^{\alpha _{1}(x)})\). So there exists \(\delta >0\) such that, for \(mE<\delta \),
From the Vitali theorem, \(\int _{\varOmega }f(x,u_{n})u_{n} \phi _{\varepsilon }\,dx\rightarrow \int _{\varOmega }f(x,u)u\phi _{\varepsilon }\,dx\). In the same way, \(\int _{\partial \varOmega }b(x, u_{n})u_{n} \phi _{\varepsilon }\,dx\rightarrow \int _{\partial \varOmega }b(x, u)u \phi _{\varepsilon }\,dx\). Then
Note that \(u_{n}\rightarrow u\) strongly in \(L^{p(x)}(B_{2\varepsilon }(x_{j}))\), thus, as \(n\rightarrow \infty \), \(\Vert \nabla \phi _{\varepsilon }\cdot u_{n} \Vert _{L^{p(x)}}\rightarrow \Vert \nabla \phi _{\varepsilon }\cdot u \Vert _{L^{p(x)}}\). Then
Note that
and
From absolute continuity of the integral, we have \(\int _{B_{2\varepsilon }(x_{j})\cap \varOmega }( \vert u \vert ^{p(x)})^{\frac{p ^{*}(x)}{p(x)}}\,dx\rightarrow 0\), then \(|\!|\!|u\cdot \nabla \phi _{\varepsilon } |\!|\!|_{L^{p(x)}}\rightarrow 0\) as \(\varepsilon \rightarrow 0\). Therefore
Similarly, we can also obtain
Thus
Similarly, by the principle of concentration compactness
Denote \(h_{2}=\sup_{x\in \varOmega } h(x)\). For any \(j\in J\), we have \(\mu _{j}\leq h_{2}\nu _{j}\). Suppose there exists \(j_{0}\in J\) such that \(\mu _{j_{0}}=\mu _{x_{j_{0}}}>0\). If \(\mu _{j_{0}} \geq 1\), then \(\nu _{j_{0}}\leq C_{*}(h_{2}\nu _{j})^{\frac{p_{x}^{*+}}{p_{x}^{-}}}\), and further
If \(\mu _{j_{0}}< 1\), then
Note that \(\int _{\varOmega }\vert u_{n}\vert ^{p^{*}(x)}\,dx\) is bounded and \(\int _{\varOmega }\vert u_{n}\vert ^{p^{*}(x)}\,dx\rightarrow \int _{\varOmega }1\,d\nu = \nu (\overline{\varOmega })\) as \(n\rightarrow \infty \), so \(\nu _{j_{0}}= \nu (\{x_{j_{0}}\})\leq \nu (\overline{\varOmega })<\infty \). Since \(p_{x}^{-}\leq p_{x}^{+} < p_{x}^{*-}\leq p_{x}^{*+}\), there exists \(M>0\) such that, for \(h_{2}\leq M\),
which is a contradiction. So there exists \(M>0\) such that, for \(h(x)\leq M\), \(\nu _{j}=0\), \(\mu _{j}=0\), where any \(j\in J\).
b. Next, we show that \(u_{n}\rightarrow u\) strongly in \(L^{p^{*(x)}}( \varOmega )\) as \(n\rightarrow \infty \). From the discussion above, we know if \(h(x)\leq M\), then \(\nu =\vert u\vert ^{p^{*}(x)}\). Thus
As \(\vert u_{n}-u\vert ^{p^{*(x)}}\leq 2^{p_{2}^{*}}(\vert u_{n}\vert ^{p^{(x)}}+\vert u\vert ^{p ^{*(x)}})\), by the Fatou lemma, we have
then \(\lim_{n\rightarrow \infty }\sup \int _{\varOmega }\vert u_{n}-u\vert ^{p ^{*}(x)}\,dx=0\), and further \(\int _{\varOmega }\vert u_{n}-u\vert ^{p^{*}(x)}\,dx \rightarrow 0\). So \(u_{n}\rightarrow u\) strongly in \(L^{p^{*}(x)}( \varOmega )\) as \(n\rightarrow \infty \). □
4 Multiple solutions for the problems
First, let us introduce some notation. Let \(O(N)\) be the group of orthogonal linear transformations in \(\mathbb{R}^{N}\), and G be a subgroup of \(O(N)\). For \(x\neq 0\), we denote the cardinality of \(G_{x}=\{gx:g\in G\} \) by \(\vert G_{x}\vert \) and set \(\vert G\vert =\inf_{x\in \bar{ \varOmega ,x\neq 0}}\vert G_{x}\vert \). An open subset Ω of \(\mathbb{R}^{N}\) is G-invariant if \(g\varOmega =\varOmega \) for any \(g\in G\).
Definition 4.1
Let Ω be a G-invariant open subset of \(\mathbb{R}^{N}\). The action of G on \(W^{1,p(x)}(\varOmega )\) is defined \(gu(x)=u(g^{-1}x)\) for any \(u\in W^{1,p(x)}(\varOmega )\). The subspace of invariant functions is defined by
A functional \(\varphi : W^{1,p(x)}(\varOmega )\rightarrow \mathbb{R}\) is G-invariant if \(\varphi \circ g=\varphi \) for any \(g\in G\).
If the space X is a separable and reflexive Banach space, there exist \(\{e_{n}\}_{n=1}^{\infty }\subset X\) and \(\{f_{n=1}^{\infty }\}\subset X^{*}\) such that
and
For \(k=1,2,\ldots \) we denote
In order to obtain the multiple solutions for the equation, we need the following hypotheses.
Let Ω be a G-invariant subset of \(\mathbb{R}^{N}\), \(p(x)\) is Lipschitz continuous and G-invariant, and it satisfies \(1< p_{1}\leq p(x) \leq p_{2}< N\). We have:
- (F5):
-
\(f(gx,t)=f(x,t)\) for any \(g\in G\), \(x\in \varOmega \), \(t\in \mathbb{R}\).
- (B5):
-
\(b(gx,t)=b(x,t)\) for any \(g\in G\), \(x\in \varOmega \), \(t\in \mathbb{R}\).
- (A6):
-
\(A(x,\nabla gu)=A(x,\nabla u)\) for any \(g\in G\), \(x\in \varOmega \).
In the following, denote \(G=O(N)\). It is immediate that \(I(u)\in C ^{1}(X,\mathbb{R})\) is G-invariant. Then, by the principle of symmetric criticality, we know that u is a critical point of I if and only if u is a critical point of \(I\vert _{W_{G}^{1,p(x)}}\). Therefore, it suffices to prove the existence of a sequence of critical points for I on \(W_{G}^{1,p(x)}\).
In the following, we prove the existence of a sequence of critical points for I by the fountain theorem, and we take \(X=W^{1,p(x)}_{G}( {\varOmega })\).
Lemma 4.1
([21], Lemma 3.3)
For any \(x\in \bar{\varOmega }\), denote \(\psi _{k}=\sup_{u\in Z_{k},|\!|\!|u|\!|\!|=1}\int _{\varOmega }\vert u\vert ^{p ^{*}(x)}\,dx\), then \(\lim_{k\rightarrow \infty }\psi _{k}=0\).
Lemma 4.2
If \(\alpha (x)\in C(\bar{\varOmega })\), \(\alpha (x)>1\) and \(\alpha (x) \ll p_{*}(x)\) for any \(x\in \bar{\varOmega }\), denote \(\gamma _{k}=\sup_{u \in Z_{k},|\!|\!|u|\!|\!|=1}\int _{\partial \varOmega }\vert u\vert ^{\alpha (x)}\,dx\), then \(\lim_{k\rightarrow \infty } \gamma _{k}=0\).
Proof
Because \(0<\gamma _{k+1}\leq {\gamma _{k}}\), \(\gamma _{k}\rightarrow \gamma \geq 0\), there exists \(u_{k}\in Z_{k}\) such that \(|\!|\!|u_{k}|\!|\!|=1\) and
As \(W^{1,p(x)}_{G}{(\varOmega )}\) is reflexive, passing to a subsequence (still denoted by \(\{u_{k}\}\)), we may assume that there exists \(u\in W^{1,p(x)}_{G}{(\varOmega )}\) such that \(u_{k}\rightarrow u\) weakly in \(W^{1,p(x)}_{G}{(\varOmega )}\). For any \(f_{m}\in \{f_{n},n=1,2,\ldots \}\), we have \(f_{m}(u_{k})=0\) when \(m< k\), then \(\lim_{k \rightarrow \infty } f_{m}(u_{k})=f_{m}(u)=0\). So for any \(m\in N\), \(f_{m}(u)=0\), which implies that \(u=0\), and further \(u_{k}\rightarrow 0\) weakly in \(W^{1,p(x)}_{G}{(\varOmega )}\). According to Theorem 2.6, the embedding \(W^{1,p(x)}(\varOmega )\rightarrow L^{\alpha (x)}(\partial \varOmega )\) is compact, so \(u_{k}\rightarrow 0\) strongly in \(L^{\alpha (x)}{(\partial \varOmega )}\), that is, \(\Vert u_{k}\Vert _{L^{\alpha (x)}(\partial \varOmega )}\to 0\). Thus \(\gamma _{k}\to 0\) as \(k\to \infty \). □
Theorem 4.1
Assume hypotheses (F1), (F2), (F3) and (F5) or (F̃1), (F2), (F5), (B̃1), (B2), (B3), (B5) and (H1) are fulfilled, \(p(x)\) is a Lipschitz continuous function on Ω̄ and G-invariant. Then there exists \(M>0\) such that, whenever \(h(x)\leq M\), the problem has a sequence of weak solutions \(\{u_{n}\}\) such that \(I(u_{n})\to \infty \), as \(n\to \infty \).
The theorem will be verified by the fountain theorem in three steps.
Proof
(1) For every \(k \in N\), there exists \(\gamma _{k}>0\), such that \(\inf_{u \in Z_{k} ,|\!|\!|u |\!|\!|= \gamma _{k} } I(u) \to \infty \) as \(k \to \infty \).
From (F̃1) and (B̃1)
then
By the Young inequalities
then
Denote
As \(\alpha _{2} (x) \ll p_{*} (x) \), so by Lemma 4.2 and Lemma 4.1, we obtain \(\omega _{k} \to 0\) and \(\beta _{k} \to 0\) as \(k \to \infty \). Take
Then, for \(|\!|\!|u |\!|\!|> 1\), we have
From the Young inequality
then
Next, we consider the following equation:
Let \(t_{k}\) be the solution of (4.2),
\(t_{k} \to \infty \) as \(k \to \infty \). We choose \(\gamma _{k} = t_{k}\), thus, for \(|\!|\!|u |\!|\!|= \gamma _{k}\), \(k \to \infty \), we have
(2) For all \(k \in N\), there exists \(\rho _{k} > \gamma _{k}\) such that \(\max_{u \in Y_{k} , |\!|\!|u|\!|\!|= \rho _{k} } I(u) \le 0\) as \(k \to \infty \), where \(\gamma _{k}\) is given by (1).
From (4.1), we have
By the Young inequality
We choose \(\varepsilon _{1} = \min \{ 1,\frac{h_{1} }{4C_{1} p _{2}^{*} } \}\), \(\varepsilon _{2} = \min \{ 1,\frac{\alpha _{1}^{-} h_{1}}{4C_{1} p_{2}^{*} } \}\), then \(C_{1} \varepsilon _{1} \le \frac{h_{1} }{4p_{2}^{*} }\), \(\frac{C_{1} \varepsilon _{2} }{ \alpha _{1}^{-} } \le \frac{h_{1} }{4p_{2}^{*} }\). Thus
As \(p(x)\), \(p^{*}(x)\) are continuous on Ω̄, and \(p(x) \ll p^{*} (x)\). Similarly to Theorem 3.1 we can get hypercubes \(\{\varOmega _{i}\}_{i=1}^{m}\) which mutually have no common points and \(\overline{\varOmega }= \bigcup_{i = 1}^{m} {\overline{\varOmega _{i} } }\). On \(\varOmega _{i}\),
then
Since \(p(x) > 1\), from the continuous embedding \(L ^{p(x)} (\varOmega ) \to L ^{1} (\varOmega )\), there exists \(C>0\) such that
Because \(Y_{k}\) is a finite dimensional space, \(|\!|\!|u |\!|\!|\) and \(\Vert u \Vert _{L^{p^{*} (x)} }\) are equivalent. Thus, for any \(i \in \{ 1,2,\ldots,m\}\), \(|\!|\!|u |\!|\!|_{\varOmega _{i} } \ge 1\),
Let
Due to (4.3), there exist \(M_{i}>0\), \(g_{i} (t)\) negative and monotone decreasing for any \(t\in [M_{i}, +\infty )\), and \(g_{i}(t) \to -\infty \) as \(t\to \infty \). Denote \(t_{0} =\max \{1,M_{i}, i=1,2,\ldots,m\}\), when \(t>t_{0}\), we have \(g_{j} (t)\leq 0\) for \(j\in \{i=1,2,\ldots,m\}\).
For any \(i\in \{1,2,\ldots,m\}\), \(|\!|\!|u|\!|\!|_{\varOmega _{i}}\geq t_{0}\) when \(|\!|\!|u|\!|\!|_{\varOmega _{i}}\) sufficiently large. It is easy to check that \(I(u)\leq 0\). So when \(|\!|\!|u|\!|\!|\) is large enough, we can find that \(|\!|\!|u|\!|\!|_{\varOmega _{i}}\) is sufficiently large for any \(i\in \{1,2,\ldots,m\}\). Thus \(I(u)\leq 0\) when \(|\!|\!|u|\!|\!|=\rho _{k}>\gamma _{k}\).
(3) The functional I satisfies the (PS) condition.
If the function \(f(x)\) satisfies (F1), (F2), (F3) and (F5), the proof is similar to (3) of Theorem 3.1. We only need to change the space \(W^{1,p(x)}\) to \(W^{1,p(x)}_{G}\).
If the function \(f(x)\) satisfies (F̃1), (F2) and (F5), we choose
in the proof of (3) of Theorem 3.1. Then
From (4.1) and (F̃1), we obtain
where \(\alpha _{1}^{+} = \sup_{x \in \varOmega } \alpha _{1} (x)\), \(\alpha _{1}^{-} = \inf_{x \in \varOmega } \alpha _{1} (x)\).
Since \(\alpha _{1} (x) \ll p(x)\), by the Young inequality, we have
Then
Similarly to (3) of Theorem 3.1, we find that \(\{u_{n}\}\) is bounded. By Lemma 3.5, we find that the functional I satisfies the (PS) condition.
From the fountain theorem, the proof of Theorem 4.1 follows immediately from (1), (2) and (3). □
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The authors are very grateful to the referees for carefully reading of the paper and for their useful comments and suggestions, which have improved the paper.
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This work was supported by the National Natural Science Foundation of China (Grant No. 11771107 and No. 11801120), and the Research Funding of Heilongjiang Province (Grant No. RCCX201716).
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Shan, Y., Fu, Y. Nonlinear boundary value problems of a class of elliptic equations involving critical variable exponents. Bound Value Probl 2019, 122 (2019). https://doi.org/10.1186/s13661-019-1231-z
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DOI: https://doi.org/10.1186/s13661-019-1231-z