Existence results for some anisotropic Dirichlet problems

https://doi.org/10.1016/j.jmaa.2020.124044Get rights and content

Abstract

This paper concerns with a class of elliptic anisotropic Dirichlet problems depending of one real parameter on bounded Euclidean domains. Our approach is based on variational and topological methods. More concretely, along the paper we show the existence of at least two weak solutions for the treated problem by using a direct consequence of the celebrated Pucci and Serrin theorem in addition to a local minimum result for differentiable functionals due to Ricceri. This abstract approach has been developed for equations on Carnot groups; see [15].

Introduction

It is well-known that a great attention in the last years has been focused by many authors on the study of anisotropic equations on bounded Euclidean domains. See, among others, the papers [2], [3], [4], [5], [6], [9], [11], [13] as well as [14], [17], [18], [19], [22], [23], [28] and references therein.

Motivated by this large interest in the current literature, we study here the existence of weak solutions for the following anisotropic Dirichlet problem(Pλf){Δp(x)u=λf(x,u)in Ωu=0on Ω, where Δp(x)u:=div(|p(x)2u) denotes the p()-Laplace operator, ΩIRN is a bounded domain with smooth boundary ∂Ω, λ is a positive real parameter, and p:Ω¯IR is a continuous function satisfying1<p:=infxΩp(x)p(x)p+:=supxΩp(x)<+. Moreover, f:Ω¯×IRIR is a continuous function such that

  • (f1)

    there exist a1,a2>0 and qC(Ω¯) with 1<p(x)<p for each xΩ¯, such that|f(x,t)|a1+a2|t|q(x)1, for each (x,t)Ω×IR, wherep(x):={Np(x)Np(x)if p(x)<Nif p(x)N.

Inspired by [1], [15], [24], we prove that, for small values of λ, problem (Pλf) admits at least two weak solutions [see Theorem 3.1] requiring that the continuous and subcritical nonlinear term f satisfies the celebrated Ambrosetti-Rabinowitz condition without the usual additional assumption at zero, that islimt0f(x,t)t=0, uniformly in Ω¯.

A special case of our result reads as follows.

Theorem 1.1

Let Ω be a smooth and bounded domain of the Euclidean space IRN, p>1, and f:IRIR be a continuous function for which

  • (f1)

    there exist a1,a2>0 and q(2,2(N(p1)+pNp)) such that|f(t)|a1+a2|t|q1, for every tIR;

  • (f2)

    there are μ>2 and r>0 such that0<μ0tf(τ)dτtf(t), for any |t|r.

Then, there exists an open interval Λ(0,+) such that, for every λΛ, the following problem{Δp=λf(u)inΩu=0,onΩ admits at least two (distinct) weak solutions in the Sobolev W01,p(Ω).

The interval Λ in the above result can be explicitly determined. Setcs:=supuW01,p(x)(Ω){0}uLs(Ω)uW01,p(x)(Ω),(withs{1,q})

Our abstract tool for proving the main result is the following abstract theorem that we recall here in a convenient form.

Theorem 1.2

Let E be a reflexive real Banach space, and let Φ,Ψ:EIR be two continuously Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous and coercive. Further, assume that Ψ is sequentially weakly continuous. In addition, assume that, for each α>0, the functional Jα:=αΦΨ satisfies the classical compactness Palais-Smale (briefly (PS)) condition. Then, for each ϱ>infEΦ and eachα>infuΦ1((,ϱ))supvΦ1((,ϱ))Ψ(v)Ψ(u)ϱΦ(u), the following alternative holds: either the functional Jα has a strict global minimum which lies in Φ1((,ϱ)), or Jα has at least two critical points one of which lies in Φ1((,ϱ)).

The above critical point result comes out from a joint application of the classical Pucci-Serrin theorem (see [20]) and a local minimum result due to Ricceri (see [25]). For a proof of Theorem 1.2 see, for instance, [24, Theorem 6]. We refer the interested reader to [16], [26], [27] and references therein for recent applications of the Ricceri's variational principle.

The plan of the paper is as follows. Section 2 is devoted to our abstract framework and preliminaries. Successively, in Section 3, Theorem 3.1 and some preparatory results concerning the compactness Palais-Smale condition (see Lemma 3.2, Lemma 3.3) are presented.

In the last section, Theorem 3.1 has been proved and a concrete example of an application is presented in Example 4.3.

Section snippets

Abstract framework

Here and in the sequel, we assume that pC(Ω¯) verifies the previous condition and is globally log-Hölder continuous on Ω. The variable exponent Lebesgue space Lp(x)(Ω) is definedLp(x)(Ω):={u:ΩIR:u is measurable and ρp(u):=Ω|u|p(x)dx<+}.

On Lp(x)(Ω) we consider the Luxemburg normuLp(x)(Ω):=inf{λ>0:Ω|uλ|p(x)dx1}. The generalized Lebesgue-Sobolev space W1,p(x)(Ω) is defined by puttingW1,p(x)(Ω):={uLp(x)(Ω):|u|Lp(x)(Ω)} and it is endowed with the following normuW1,p(Ω):=uLp(x)(Ω)+|u

The main result and some technical lemmas

The aim of this section is to prove that, under natural assumptions on the nonlinear term f, weak solutions to problem (Pλf) below do exist. With the above notations the main result reads as follows.

Theorem 3.1

Let f:Ω¯×IRIR be a continuous function such that

  • (f1)

    there exist a1,a2>0 and qC(Ω¯) with 1<q(x)<p for each xΩ¯, such that|f(x,t)|a1+a2|t|q(x)1, for each (x,t)Ω×IR, wherep(x):={Np(x)Np(x)ifp(x)<Nifp(x)N;

  • (f2)

    there are μ>p+ and r>0 such that0<μ0tf(x,τ)dτtf(x,t), for any xΩ¯, and |t|r.

Then, for

Proof of Theorem 3.1

For the proof of our result, we observe that problem (Pλf) has a variational structure, indeed it is the Euler-Lagrange equation of the functional Jλ. Note that the functional Jλ is continuously Gâteaux differentiable in uW01,p(x)(Ω) and one hasJλ(u),v=1λΩ|u|p(x)2u,vdxΩf(x,u)vdx for every vW01,p(x)(Ω). Thus, the critical points of Jλ are exactly the weak solutions to problem (Pλf). Let ϱ>0 and set α:=1/λ, with λ as in the statement.

Hence, let us apply Theorem 1.2, taking E:=W01,p(x

References (32)

  • G. Anello

    A note on a problem by Ricceri on the Ambrosetti-Rabinowitz condition

    Proc. Am. Math. Soc.

    (2007)
  • R. Arora et al.

    A Picone identity for variable exponent operators and applications

    Adv. Nonlinear Anal.

    (2020)
  • G. Bonanno et al.

    Existence results of infinitely many solutions for p(x)-Laplacian elliptic Dirichlet problems

    Complex Var. Elliptic Equ.

    (2012)
  • X.L. Fan et al.

    Existence and multiplicity of solutions for p(x)-Laplacian equations in IRN

    Nonlinear Anal.

    (2003)
  • X.L. Fan et al.

    Existence of solutions for p(x)-Laplacian Dirichlet problem

    Nonlinear Anal.

    (2004)
  • X.L. Fan et al.

    On the generalized Orlicz-Sobolev space Wk,p(x)(Ω)

    J. Gansu Sci.

    (1998)
  • Cited by (0)

    View full text