Existence of solutions for a nonhomogeneous Dirichlet problem involving \(p(x)\)Laplacian operator and indefinite weight
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Abstract
Keywords
Generalized Lebesgue–Sobolev spaces \(p(x)\)Laplacian operator Symmetric mountain pass lemmaMSC
35B38 35J20 35J60 35J661 Introduction
The remainder of this paper is organized as follows. In Section 2, we introduce some technical results and formulate the required hypotheses on (1.1). Sections 3 and 4 are devoted to the statement of our main results along with some auxiliary results and to their proofs. In the appendix we present the appropriate version of the mountain pass theorem related to our problem.
2 Preliminaries and hypotheses
In order to study problem (1.1), some of the properties on variable exponent Lebesgue spaces and Sobolev spaces, \(L^{p(x)}(\varOmega )\) and \(W^{1,p(x)} (\varOmega )\) respectively, are required and listed below. We refer to [15, 23, 26] for exhaustive details on properties of those spaces.
Definition 2.1
Throughout this paper, \(L^{p(x)}(\varOmega ) \) will be endowed with this norm.
Remark 2.2
Variable exponent Lebesgue spaces have many properties similar to those of classical Lebesgue spaces, namely they are separable Banach spaces and the Hölder inequality holds. The inclusions between Lebesgue spaces are also naturally generalized, that is, if \(0< \operatorname{mes}(\varOmega )<\infty \) and p, q are variable exponents such that \(p(x)< q(x)\) a.e. in Ω, then there exists a continuous embedding \(L^{q(x)}(\varOmega ) \hookrightarrow L^{p(x)}(\varOmega )\).
We recall below some statements whose details can be found in [18, 20, 23].
Let us denote by \(L^{p'(x)}(\varOmega )\) the conjugate space of \(L^{p(x)}(\varOmega )\), with \(\frac{1}{p(x)}+\frac{1}{p'(x)}=1\). There is a counterpart of the Hölder inequality for variable exponent Lebesgue spaces when \(p \in L_{+}^{\infty }(\varOmega )\) in the literature (cf. [18]). We give below a version relevant to the need of this work.
Proposition 2.3
(Hölder inequality)
Proposition 2.4
It is worth noticing that this relation between the norm and the modular shows an equivalence between the topology defined by the norm and that defined by the modular.
Proposition 2.5
Definition 2.6
Proposition 2.7
\(L^{p(x)}(\varOmega )\)and\(W^{1,p(x)}(\varOmega )\)are separable Banach spaces when\(p \in L_{+}^{\infty }(\varOmega )\), reflexive and uniformly convex for\(p \in L^{\infty }(\varOmega )\)and\(\operatorname{ess}\inf_{\varOmega } p(x) > 1 \).
Definition 2.8
We also define the space \(W_{0}^{1,p(x)}(\varOmega )\) as the closure of the space \(C_{0}^{\infty }(\varOmega )\) (\(C^{\infty }\)functions with compact support in Ω) in the space \(W^{1,p(x)}(\varOmega )\) with respect to the norm \(\ u \_{1,p(x)}\).
The dual space of \(W_{0}^{1,p(x)}(\varOmega )\) is denoted by \(W^{1,p'(x)}( \varOmega )\), where \(\frac{1}{p(x)}+\frac{1}{p'(x)}=1\), for every \(x\in \overline{\varOmega }\).
With respect to those spaces, we recall from [15, 23] the following.
Proposition 2.9
 (i)
\(W_{0}^{1,p(x)}(\varOmega )\)is a separable Banach space when\(p \in L_{+}^{\infty }(\varOmega )\), reflexive and uniformly convex when\(p \in L^{\infty }(\varOmega )\)and\(\operatorname{ess}\inf_{\varOmega } p(x) > 1 \).
Assume that\(p,q\in C_{+}(\overline{\varOmega })\). Then
 (ii)if\(p(x) < N \)and\(q(x)< p^{*}(x) = \frac{Np(x)}{Np(x)}\)for every\(x\in \overline{\varOmega }\), then there is a compact and continuous embedding$$ W^{1,p(x)}(\varOmega )\hookrightarrow L^{q(x)}(\varOmega ); $$
 (iii)if\(p(x) > N\)for every\(x\in \overline{\varOmega }\), then$$ W^{1,p(x)(\varOmega )\hookrightarrow L^{\infty }(\varOmega )}; $$
 (iv)there is a constant \(C>0\) such that$$ \vert u \vert _{p()}\leq C \Vert \nabla u \Vert _{p()},\quad \textit{for all }u\in W_{0}^{1,p(x)}( \varOmega ). $$
Remark 2.10
Using estimate (iv) of Proposition 2.9, we derive that the norm \(\ u \_{1,p(\cdot)}=\\nabla u\_{p(\cdot)}+ u _{p(\cdot)}\) is equivalent to the norm \(\ u \=\\nabla u\_{p(\cdot)}\) in \(W_{0}^{1,p(x)}(\varOmega )\). Here and henceforth, we will consider the space \(W_{0}^{1,p(x)}( \varOmega )\) equipped with the norm \(\ u \=\\nabla u\_{p(\cdot)}\).
Moreover, one can prove (cf. [15]) that the norm \(\ u \\) is weakly sequentially lower semicontinuous.
Remark 2.11
If \(q\in C_{+}(\overline{\varOmega })\) and \(q(x)< p^{*}(x)\) for every \(x\in \overline{\varOmega }\), then the embedding of \(W_{0}^{1,p(x)}( \varOmega )\) into \(L^{q(x)}(\varOmega )\) is compact.
We have the following properties (cf. [10, 22]).
Proposition 2.12
 (i)
\(\Delta _{p(x)}:W_{0}^{1,p(x)}(\varOmega )\to W^{1,p'(x)}( \varOmega )\)is a homeomorphism.
 (ii)\(\Delta _{p(x)}:W_{0}^{1,p(x)}(\varOmega )\to W^{1,p'(x)}( \varOmega )\)is a strictly monotone operator, that is,$$ \langle \Delta _{p(x)}u\Delta _{p(x)}v,uv \rangle >0\quad \textit{for all } u\neq v \in W_{0}^{1,p(x)}(\varOmega ). $$
 (iii)\(\Delta _{p(x)}:W_{0}^{1,p(x)}(\varOmega )\to W^{1,p'(x)}( \varOmega )\)is a mapping of type\((S_{+})\), that is,then\(u_{n}\to u\)in\(W_{0}^{1,p(x)}(\varOmega )\).$$ \textit{if } u_{n} \rightharpoonup u\quad \textit{in }W_{0}^{1,p(x)}( \varOmega )\quad \textit{and}\quad \limsup_{n\to \infty }\langle \Delta _{p(x)}u_{n},u _{n}u \rangle \leq 0, $$
Proposition 2.13
Note that, if \(f:\varOmega \times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function and \(u\in M\), then the function \(N_{f}u: \varOmega \to \mathbb{R}\) defined by \((N_{f}u)(x)=f(x,u(x))\) for \(x\in \varOmega \) is measurable in Ω. Thus, the Carathéodory function \(f:\varOmega \times \mathbb{R}\to \mathbb{R}\) generates an operator \(N_{f}:M\to M\), which is called the Nemytskii operator. The properties of \(N_{f}\) are recalled through the propositions below (see [35] for details).
Proposition 2.14
Proposition 2.15
 (i)F is a Carathéodory function and there exist a constant \(c_{1}\geq 0\) and \(\sigma \in L^{1}(\varOmega )\) such that$$ \bigl\vert F(x,t) \bigr\vert \leq c_{1} \vert t \vert ^{\alpha (x)}+\sigma (x)\quad \textit{for all }x \in \varOmega , t\in \mathbb{R}. $$
 (ii)
The functional\(\varPhi :L^{\alpha (x)}(\varOmega )\to \mathbb{R}\)defined by\(\varPhi (u)=\int _{\varOmega }F(x,u(x))\,dx\)is continuously Fréchet differentiable and\(\varPhi '(u)=N_{f}(u) \)for all\(u\in L^{\alpha (x)}(\varOmega )\).
Remark 2.16
 (A1)
 \(p, q, s \text{ and } \beta \in C_{+}(\overline{\varOmega })\) such that, for all \(x\in {\overline{\varOmega }}\),$$ q(x) < p(x) \leq N ,\qquad \beta (x) < p(x) \leq N, \qquad N < s(x)\quad \text{for } V\in L^{s(x)}(\varOmega ). $$(2.1)
 (A1′)
 Moreover, we assume that$$ q^{+}_{\varOmega _{}}= \sup_{\varOmega _{}}q(x) < p^{}\quad \text{and} \quad \beta ^{+} < p^{} . $$(2.2)
 (A2)
 where \(c\geq 0\) is constant, \(\beta \in C_{+}(\overline{\varOmega })\) with \(\beta (x)< p^{*}(x)\) for every \(x\in \overline{\varOmega }\), and \(h\in L^{\infty }(\varOmega ) \).$$ \bigl\vert f(x,t) \bigr\vert \leq c \vert t \vert ^{\beta (x)1}+h(x)\quad \text{for all }x\in \varOmega , t\in \mathbb{R}, $$(2.3)
 (A3)
 There exist \(\gamma \in L^{\infty }(\varOmega )\), \(\gamma > 0 \), and θ satisfying \(\theta < p^{}\) such thatwhere \(F(x,s)=\int _{0}^{s}f(x,t)\,dt\).$$ F(x,t)\geq \gamma (x) \vert t \vert ^{\theta }\quad \text{for all }x\in \varOmega \text{ and any } t \in [0,1[, $$(2.4)
 (A4)

\(f(x,t)=f(x,t)\) for \(x\in \varOmega \), \(t\in \mathbb{R}\).
3 Main results
From assumptions (A1) on the functions p, q, s and from (3.1), a straightforward computation gives \(q(x)< p^{*}(x)\), \(\beta (x) < p^{*}(x)\), \(s'(x)q(x) < p^{*}(x)\), \(\alpha (x) < p^{*}(x) \) for every \(x \in \overline{\varOmega }\). Hence, we have the following remark.
Remark 3.1
The functional H is obviously well defined and satisfies the following.
Proposition 3.2
The functionalHis continuously Fréchet differentiable and is weakly lower semicontinuous. Moreover, \(u \in X\)is a critical point ofHif and only ifuis a weak solution of (1.1).
Proof
Let us show that H is sequentially weakly lower semicontinuous.
Next, we have the following.
Proposition 3.3
Under assumptions (A1), (A1′), (A2), the energy functional (H) is coercive.
Proof
We are now ready to state the first existence result of this work.
Theorem 3.4
Under assumptions (A1), (A1′), and (A2), problem (1.1) has a weak solution.
Proof
Since H is differentiable, coercive, weakly lower semicontinuous, H has a critical minimum point u in X which is a weak solution of (1.1) and then the proof is complete. □
Remark 3.5
Notice that when V is positive, the first condition in (A1′) is not necessary and the proof of the coercivity of H is of course trivial.
Corollary 3.6
Suppose that hypotheses of Theorem 3.4are satisfied. If besides\(V\in L^{s(x)}(\varOmega )\)is a nonnegative function and\(f(x,\cdot ): \mathbb{R}\to \mathbb{R}\)defined by\(f(x,\cdot )(t)=f(x,t)\)is a decreasing function for a.e. \(x\in \varOmega \), then problem (1.1) has a unique weak solution inX.
Proof
Remark 3.7
In addition to the assumptions in Corollary 3.6, if the Carathéodory function \(f:\varOmega \times \mathbb{R}\to \mathbb{R}\) satisfies (A4), then \(u\equiv 0\) is a trivial solution, and since the solution is unique when it exists, then \(u\equiv 0\) is the only one solution in X for Dirichlet problem (1.1).
In the next step, we will suppose that condition (A1′) is no longer satisfied and of course the coercivity of the functional H fails since V is a signchanging function. In this case, we prove that H satisfies the Palais–Smale (PS) condition and show multiplicity results for our problem.
Definition 3.8
The \(C^{1}\)functional H is said to satisfy the Palais–Smale condition, in short the (PS) condition, if any sequence \((u_{n})_{n\in \mathbb{N}}\subseteq X\), for which \((H(u_{n}))_{n \in \mathbb{N}}\subseteq \mathbb{R}\) is bounded and \(dH(u_{n})\to 0\) as \(n\to \infty \), has a convergent subsequence.
Proposition 3.9
Under assumptions (A1) and (A2), the functionalHsatisfies the (PS) condition.
The following lemma plays a key role in the proof of Proposition (3.9). The constant exponent version (\(p(x) \equiv p\)) of the lemma can be found in [14].
Lemma 3.10
Remark 3.11
Clearly, in Lemma (3.10), \(\int _{\varOmega }\omega u^{q(x)} \,dx < \infty \) for any \(u \in X\). The case \(r(x)= 1 \) with \(q (x) >N \) is obvious since \(p (x) >N \) and then \(X\hookrightarrow L^{\infty }(\varOmega )\).
Proof of Lemma 3.10
A similar approach enables us to get (3.13).
Proof of Proposition 3.9
We are now ready to state the following existence result for multiple solutions.
Theorem 3.12
Under assumptions (A1), (A2), (A3), and (A4), problem (1.1) has a bounded sequence of weak solutions\((u_{n})_{n\in \mathbb{N}}\subseteq X\)such that\(H(u_{n}) = c_{n} < 0 \). Moreover, the sequence of the critical values\((c_{n})_{n}\)tends to\(c= \inf_{X}H\).
Theorem 3.12 is derived by searching solutions as critical values of the functional H by means of a symmetric version of the mountain pass theorem.
Many versions of the mountain pass theorem stated according to the geometry of the problem under consideration exist in the literature (see [3, 31, 33, 34]). We state below an appropriate version to our situation.
Theorem 3.13
(A symmetric mountain pass theorem )
 (i)
there are some constants\(\rho ,\alpha >0\)such that\(H(u)\geq \alpha \)for all\(u\in X\)with\(\ u\\geq \rho \);
 (ii)
for each finite dimensional subspaceFofX, there is a constant\(R > 0\)such that\(H(u)< 0\)for all\(u\in F\)with\(\ u\ \leq R\).
The proof of Theorem 3.13 can be adapted from some works in the literature. We give it as an appendix to this work for the sake of completeness.
4 Proof of Theorem 3.12 and auxiliary results
The proof will consist in showing that H satisfies the geometry required to apply the symmetric mountain pass theorem.
Lemma 4.1
Under assumptions (A1), (A2), and (A3), there exist some constants\(\rho ,\alpha > 0\)such that\(H_{ \{u\in X: \lVert u\rVert \geq \rho \} }\geq \alpha \).
Proof
Lemma 4.2
Suppose that assumptions (A1) and (A3) are satisfied. Then, for each finite dimensional subspaceF, there is a constant\(R>0\)such that\(H(u)\leq 0\)for all\(u\in F\)with\(\ u\ \leq R\).
Proof
To conclude with the proof of Theorem 3.12, we notice from (A4) that \(H(0) = 0 \) and H is even, and from Lemmas 4.1, 4.2, H satisfies the conditions required in Theorem 3.12, and then the result is achieved.
An auxiliary result in terms of multiple solutions in the same spirit of the works in [1, 6], and [22] can be obtained in our context by means of the fountain theorem. However, the sequence of solutions obtained either by the mountain pass theorem or the fountain theorem are quite different. Of course, in our context, the sequence of critical values obtained via the mountain pass theorem converges to a nonzero limit, while the use of the fountain theorem gives rise to critical values sequence converging to 0 as stated in Theorem 4.3 below.
Theorem 4.3
(Dual fountain theorem)
Suppose thatXis an infinite dimensional reflexive separable Banach space. Let\(H \in C^{1}(X,\mathbb{R})\)be even and satisfy the (PS) condition and\(H(0)=0\).
 (i)
\(H(u)\geq 0\)for all\(u\in Z_{k}\)with\(\ u\=\rho _{k}\),
 (ii)
\(H(u)< 0\)for all\(u\in Y_{k} \)with\(\ u\ = r_{k}\),
 (iii)
\(d_{k} = \inf_{\{u\in Z_{k}, \u\ \leq \rho _{k}\}}H(u) \to 0 \)as\(k \to +\infty \).
Sketch of the proof
H satisfies of course (PS), and by means of Lemmas 4.1 and 4.2, (i), (ii) are also satisfied. Thus, to prove Theorem 4.3, we need only to show that (iii) is satisfied. Accordingly, we need the following lemma whose proof can be pointed out similarly as in [1, 6], and [22].
Lemma 4.4
Notes
Acknowledgements
The authors wish to thank the referees for their interesting remarks and suggestions which contributed to the quality of the paper.
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