Abstract
In this paper, a nonlinear elliptic obstacle problem is studied. The nonlinear nonhomogeneous partial differential operator generalizes the notions of p-Laplacian while on the right hand side we have a multivalued convection term (i.e., a multivalued reaction term may depend also on the gradient of the solution). The main result of the paper provides existence of the solutions as well as bondedness and closedness of the set of weak solutions of the problem, under quite general assumptions on the data. The main tool of the paper is the surjectivity theorem for multivalued functions given by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone one.
Highlights
Let ⊆ RN be a bounded domain with a Lipschitz-boundary ∂
For the nonlinear elliptic problems with gradient dependence we refer to the following papers: Averna-Motreanu-Tornatore [1], Bai [2], Bai-Gasinski-Papageorgiou [3], Faraci-Motreanu-Puglisi [4], Gasinski-Papageorgiou [5,6], Gasinski-Winkert [7], Motreanu-Motreanu-Moussaoui [8], Guarnotta-Marano-Motreanu [9], PapageorgiouRadulescu-Repovš [10], Faraci-Puglisi [11], Figueiredo-Madeira [12], PapageorgiouRadulescu-Repovš [13], Tanaka [14], Guarnotta-Marano [15], Liu-Motreanu-Zeng [16], Marano-Winkert [17], Araujo-Faria [18], Bai-Papageorgiou-Zeng [19]
The main tool in the proof of the existence result for problem (1.1) will be the surjectivity result due to Le [20] for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping
Summary
Let ⊆ RN be a bounded domain with a Lipschitz-boundary ∂. We study the following nonlinear nonhomogeneous elliptic problem with a multivalued convection term and under obstacle condition. Where a : × RN → RN is continuous, monotone with respect to the second variable and satisfies particular other growth conditions to be described later, reaction term f : × R × RN → 2R is multivalued and depends on the gradient of the solution (which makes the problem nonvariational) and obstacle : → [0, +∞] is a given function. None of the above papers deals with multivalued or obstacle problems. For problems with double phase operators (which include the case for the ( p, q)-Laplacian) and multivalued terms (and dealing with obstacle problems), we refer to Zeng-Gasinski-Winkert-Bai [23,24,25]. We mention that MingioneRadulescu [26] provided an overview of recent results concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators
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