On strong solvability of the Dirichlet problem for a class of semilinear elliptic equations with discontinuous coefficients

Abstract

In this paper, we study a solvability result for the nonlinear problemassuming for the weight functions v ∈ A ∞ , ω ∈ A p to belong the Muckenhoupt class and a balance condition of Chanillo-Wheeden's type, with degenerate gradient ∇ ω u = ω 1/p ∇ x , ∇ y and its mod-The range conditions q ∈ (p, pN/(N − p)) and γ ∈ (1, N/(N − 1)) (or γ ∈ (1, p) and v −γ/(q−γ) ∈ L 1,loc (Ω) additionally) and µ ∈ (0, Λ) with sufficiently small Λ are assumed also.