Abstract
We consider an elliptic hemivariational inequality with nonlocal nonlinearities. Assuming only certain growth conditions on the data, we are able to prove existence results for the problem under consideration. In particular, no continuity assumptions are imposed on the nonlocal term. The proofs rely on a combined use of recent results due to the authors on hemivariational inequalities and operator equations in partially ordered sets.
Highlights
Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, and let V = W1,p(Ω) and V0 = W01,p(Ω), 1 < p < ∞, denote the usual Sobolev spaces with their dual spaces V ∗ and V0∗, respectively
We deal with the following quasilinear hemivariational inequality: u ∈ V0 : − Δpu, v − u + jo(u; v − u)dx ≥ Ᏺu, v − u, ∀ v ∈ V0, Ω
While elliptic hemivariational inequalities in the form (1.1) with Ᏺu replaced by a given element f ∈ V0∗ have been treated recently, for example, in [2] under the assumption that appropriately defined super- and subsolutions are available, the novelty of the problem under consideration is that the term on the right-hand side of (1.1) is nonlocal and not necessarily continuous in u
Summary
Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, and let V = W1,p(Ω) and V0 = W01,p(Ω), 1 < p < ∞, denote the usual Sobolev spaces with their dual spaces V ∗ and V0∗, respectively. We deal with the following quasilinear hemivariational inequality:. Where jo(s;r) denotes the generalized directional derivative of the locally Lipschitz function j : R → R at s in the direction r given by jo(s; r). The mapping Ᏺ : V0 → V0∗ on the right-hand side of (1.1) comprises the nonlocal term and is generated by a function F : Ω × Lp(Ω) → R through.
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