Nonlinear nonhomogeneous Robin problems with gradient dependent reaction
Introduction
Let () 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):
In this problem the map 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 below. These hypotheses are mild and natural. This way we incorporate in our framework many differential operators of interest, such as the -Laplacian and the -Laplacian (that is, the sum of a -Laplacian and a -Laplacian). The reaction term depends also on the gradient of the unknown function . This makes the problem nonvariational and so a topological approach is necessary. In the boundary condition denotes the conormal derivative of for the differential operator, defined by extension of the map to all , with being the outward unit normal on 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 .
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 -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 . 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 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 be a reflexive Banach space and its topological dual. By we denote duality brackets for the pair . Let . The graph of is the set and the domain of is the set We say that is “monotone”, if The map is “strictly monotone”, if We say that is “maximal monotone”, if the following property holds This means that is
Existence of solutions
Consider the nonlinear map defined by
Proposition 3.1 If hypothesis holds, then is bounded (that is, maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone too) and of type .
Proof From hypothesis it is clear that is bounded, continuous, monotone (hence it is also maximal monotone). It remains to show that is of type . To this end let be a sequence such that
Uniqueness of the solution
In this section under stronger conditions on and the reaction term , we prove the uniqueness of the solution for problem (1.1). The new hypotheses are the following:
is continuous, strictly monotone, and
- (i)
there exists such that for all ;
- (ii)
there exists such that for all .
: is a measurable function such that
- (i),(ii)
are the same as corresponding hypotheses and ;
- (iii)
Acknowledgments
The authors wish to thank the two anonymous reviewers for their corrections and remarks.
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