Formula omitted] existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions

Abstract

In this paper we study the questions of existence and uniqueness of weak and entropy solutions for equations of type −diva(x,Du)+γ(u)∋ϕ, posed in an open bounded subset Ω of RN, with nonlinear boundary conditions of the form a(x,Du)⋅η+β(u)∋ψ. The nonlinear elliptic operator diva(x,Du) is modeled on the p-Laplacian operator Δp(u)=div(|Du|p−2Du), with p>1, γ and β are maximal monotone graphs in R2 such that 0∈γ(0) and 0∈β(0), and the data ϕ∈L1(Ω) and ψ∈L1(∂Ω).