Abstract
On the Existence and Uniqueness of Solutions for a Class of Nonlinear Degenerate Elliptic Problems via Browder-Minty Theorem
Highlights
The goal of this paper is to show that there is a unique weak solution in W01,p(Ω, ν1) (p is not necessarily equal to 2) for the Dirichlet problem associated with the nonlinear degenerate elliptic equation of the form:
Φ(y) = 0 on ∂Ω, where Ω is a bounded open set in RN ; ν1, ν2, and ν3 are Ap-weight functions, and the functions b : Ω × R × RN −→ RN, a : Ω × RN −→ RN and g : Ω × R −→ R are Carateodory functions that satisfy some assumptions with φ ∈ Lp (Ω, ν11−p )
A function φ ∈ W01,p(Ω, ν1) is a weak solution of (1) if for any v ∈ W01,p(Ω, ν1) it holds that a(y, ∇φ), ∇v ν1dy + b(y, φ, ∇φ), ∇v ν2dy + g(y, φ)vν3dy = φvdy
Summary
The goal of this paper is to show that there is a unique weak solution in W01,p(Ω, ν1) (p is not necessarily equal to 2) for the Dirichlet problem associated with the nonlinear degenerate elliptic equation of the form:. The problems of the type (1) have already been studied for the case ν1 ≡ ν2 ≡ ν3 ≡ 1; the existence results have been reported in [4] (see [7]) when a(y, ∇φ) = 0. The degenerate case with different conditions have been investigated in many papers; for example, see [1–3, 9, 14–23]. Definitions and some preliminary results are presented .
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