Existence of solutions to nonlocal and singular elliptic problems via Galerkin method

Abstract

We study the existence of solutions to the nonlocal elliptic equation -M (||u||2) Δu = ƒ(x, u) with zero Dirichlet boundary conditions on a bounded and smooth domain of ℝn. We consider the M-linear case with ƒ ∈ H-1 (Ω), and the sub-linear case ƒ(u) = uα, 0 < α < 1. Our main tool is the Galerkin method for both cases when M continuous and when M is discontinuous.