Existence of Solutions for a Nonlinear Dirichlet Problem Involving Gradient Dependent Lipschitz Convection Function

Teffera M. Asfaw, Anteneh T. Adimasu and Achamyelesh A. Aligaz

Journal of Mathematics, 2025, vol. 2025, 1-14

Abstract: In this paper, our goal is to prove the existence of a weak solution (in H01Ω) for a fully nonlinear Dirichlet problem with a nonmonotone (e.g., Lipschitz) convection function F that depends on ∇u, and a nonlinearity G that is not necessarily monotone and depends on the solution function u, and the higher order term is −ΔΓx,u−divax,u,∇u provided that F, a, G, and Γ are Caratheòdory functions satisfying mild growth conditions. Here, Ω is a nonempty, bounded and open subset of RN with N∈Z+. We shall accomplish our goal by proving abstract existence results on the solvability of operator inclusion problems (in a Banach space X) of the type Au+Bu+Cu∋f and ATu+Bu+Cu∋f, where f∈X, A:X⊇DA⟶2X is a β-expansive and m-accretive operator, B:X⟶X is a L-Lipschitz continuous operator, C:X⟶X is a compact operator and T:X⟶X is a β-expansive and continuous operator such that for some α>0, we have Tu−Tv,ju−v≥−αu−v2 for all u,v∈X2 and ju−v∈Ju−v. The proofs are mainly based on the recent result on surjectivity of compact perturbation of β-expansive operator due to Asfaw. The abstract results and applications are new and give improvements (and/or generalizations) of those known results.

Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:6014643

DOI: 10.1155/jom/6014643

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