On a Strongly Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces

Abstract

In this paper, we prove the existence and uniqueness of weak solution to a strongly nonlinear degenerate elliptic problem of the type:

−div|ω1a(x, ▽u) + ω2b(x, u, ▽u)| + ω3g(x)u(x) = f (x).

Here, ω1, ω2 and ω3 are Ap-weight functions that will be defined in the preliminaries, where, Ω is a bounded open set of R n (n ≥ 2) and f ∈ L1 (Ω), with b: Ω × R × R n −→ R, a: Ω × R n -> R and g: Ω -> R are functions that satisfy some conditions and f belongs to L p 0 (Ω, ω 1−p 0 1). First, we transformed the problem into an equivalent operator equation; second, we utilized the Browder-Minty Theorem to prove the existence and uniqueness of weak solution to the considered problem.