Nonlinear nonhomogeneous Robin problems with gradient dependent reaction

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Abstract

We consider a Robin problem driven by a nonlinear, nonhomogeneous differential operator and a reaction term which depends on the gradient. Using a topological approach based on the Leray–Schauder Alternative Principle, we show the existence of a weak solution.

Introduction

Let ΩRN (N2) be a bounded domain with a locally Lipschitz boundary. In this paper we examine the existence of weak solutions for the following nonlinear nonhomogeneous Robin problem with gradient dependence (convection): diva(Du(z))=f(z,u(z),Du(z))inΩ,una+β(z)|u|p2u=0onΩ.

In this problem the map a:RNRN involved in the definition of the differential operator of the problem, is a continuous monotone (hence maximal monotone too) and satisfies certain other growth conditions listed in hypotheses H(a) below. These hypotheses are mild and natural. This way we incorporate in our framework many differential operators of interest, such as the p-Laplacian and the (p,q)-Laplacian (that is, the sum of a p-Laplacian and a q-Laplacian). The reaction term f(z,x,y) depends also on the gradient of the unknown function u. This makes the problem nonvariational and so a topological approach is necessary. In the boundary condition una denotes the conormal derivative of u for the differential operator, defined by extension of the map C1(Ω¯)u(a(Du),n)RNto all uW1,p(Ω), with n being the outward unit normal on Ω according to the nonlinear Green’s identity (see Gasiński–Papageorgiou [1, p. 21]).

Using the theory of operators of monotone type, we prove the existence of solutions for problem (1.1) under general conditions on the reaction f(z,,).

Nonlinear elliptic problems with gradient dependence were studied by Gasiński–Papageorgiou [2], Kourogenis–Papageorgiou [3], Bai–Gasiński–Papageorgiou [4], Bai [5], Papageorgiou–Rădulescu–Repovš [6], Tanaka [7]. Closer to our work here are the papers of Averna–Motreanu–Tornatore [8], Gasiński–Papageorgiou [9] and Marano–Winkert [10]. In the first the authors deal with a Dirichlet (p,q)-equation and their method of proof uses the theory of pseudomonotone operators. In the second the authors consider a Neumann problem driven by a differential operator of the form div(a(u)Du). Their approach uses the method of upper and lower solutions. In the third the authors deal with nonhomogeneous Robin problems and their approach is similar to that of Averna–Motreanu–Tornatore [8] based on pseudomonotone operators. In Marano–Winkert [10] no interaction with the principal eigenvalue λ̂1(p) is allowed (nonuniform nonresonance). Finally we mention that recently Robin problems of variational structure were studied by Amster [11] and Papageorgiou–Rădulescu–Repovš [12], [13]. For other problems with gradient dependent reaction term we refer to Bai–Gasiński–Papageorgiou [14], Candito–Gasiński–Papageorgiou [15] and Gasiński–Winkert [16].

Section snippets

Preliminaries — hypotheses

Let X be a reflexive Banach space and X its topological dual. By ,, we denote duality brackets for the pair (X,X). Let A:X2X. The graph of A is the set GrA={(u,u)X×X:uA(u)}and the domain of A is the set D(A)={uX:A(u)}.We say that A is “monotone”, if uv,uv0(u,u),(v,v)GrA.The map A is “strictly monotone”, if uv,uv>0(u,u),(v,v)GrA,uv.We say that A is “maximal monotone”, if the following property holds uv,uv0(u,u)GrA(v,v)GrA.This means that GrA is

Existence of solutions

Consider the nonlinear map A:W1,p(Ω)W1,p(Ω) defined by A(u),h=Ω(a(Du),Dh)RNdzu,hW1,p(Ω).

Proposition 3.1

If hypothesis H(a) holds, then A is bounded (that is, maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone too) and of type (S)+.

Proof

From hypothesis H(a) it is clear that A is bounded, continuous, monotone (hence it is also maximal monotone).

It remains to show that A is of type (S)+. To this end let {un}n1W1,p(Ω) be a sequence such that unwuinW1,p(Ω)andlim supn+A(un),unu

Uniqueness of the solution

In this section under stronger conditions on a and the reaction term f, we prove the uniqueness of the solution for problem (1.1). The new hypotheses are the following:

H(a): a:RNRN is continuous, strictly monotone, a(0)=0 and

    (i)

    there exists c0>0 such that |a(y)|c0(1+|y|) for all yRN;

    (ii)

    there exists c1>0 such that c1|yy|(a(y)a(y),yy)RN for all y,yRN.

H(f): f:Ω×R×RNR is a measurable function such that

    (i),(ii)

    are the same as corresponding hypotheses H(f)(i) and (ii);

    (iii)

    |f(z,x,y)f

Acknowledgments

The authors wish to thank the two anonymous reviewers for their corrections and remarks.

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