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

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 −ΔΓ(x, u) − diva(x, 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, j(u − v)〉 ≥ −α‖u − v‖2 for all (u, v) ∈ X2 and j(u − v) ∈ J(u − 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.