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

Abstract

<p>We study the existence of solutions to the nonlocal elliptic equation</p> <pre>-M (||u||²) Δu = ƒ(x, u)</pre> <p>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) = u^α, 0 < α < 1. Our main tool is the Galerkin method for both cases when M continuous and when M is discontinuous.</p>