Abstract
In this work we study the existence of solutions u in W^{1,p}_0(Omega ) to the implicit elliptic problem f(x, u, nabla u, Delta _p u)= 0 in Omega , where Omega is a bounded domain in {mathbb {R}}^N , N ge 2 , with smooth boundary partial Omega , 1< p< infty , and f:Omega times {mathbb {R}}times {mathbb {R}}^N times {mathbb {R}}rightarrow {mathbb {R}}. We choose the particular case when the function f can be expressed in the form f(x, z, w, y)= varphi (x, z, w)- psi (y) , where the function psi depends only on the p-Laplacian Delta _p u . We also present some applications of our results.
Highlights
Introduction and Main ResultsLet Ω ⊂ RN, N ≥ 2, be a bounded domain with smooth boundary ∂Ω, let 1 < p < ∞, let Y ⊆ R be a nonempty interval possibly coinciding with R, and let f : Ω × R × RN × R → R
In this work we study the existence of solutions u ∈ W01,p(Ω) to the implicit elliptic problem f (x, u, ∇u, Δpu) = 0 in Ω, where Ω is a bounded domain in RN, N ≥ 2, with smooth boundary ∂Ω, 1 < p < ∞, and f : Ω × R × RN × R → R
When φ is discontinuous we essentially follow [16, Theorem 3.1] to construct an appropriate upper semicontinuous multifunction F related with ψ−1 and φ, and we solve the elliptic differential inclusion −Δpu ∈ F (x, u) using the following
Summary
Let Ω ⊂ RN , N ≥ 2, be a bounded domain with smooth boundary ∂Ω, let 1 < p < ∞, let Y ⊆ R be a nonempty interval possibly coinciding with R, and let f : Ω × R × RN × R → R. We further distinguish among the case when φ is a Caratheodory function depending on x, u, and ∇u, and the case when φ is allowed to be highly discontinuous in each variable. In this last case, the dependence on the gradient is no more allowed. The dependence on the gradient is no more allowed In both situations we first reduce problem (1.1) to an elliptic differential inclusion, but methods used are different and depend on the regularity of the function φ and on the structure of the problem
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