Existence of solutions for semilinear elliptic boundary value problems on arbitrary open sets

Abstract

We show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set Ω. We make no assumptions about the open set Ω and very mild regularity assumptions on the semilinearity f, plus a coerciveness assumption which depends on the optimal Poincar´e-Steklov constant λ1. The proof is based on Schaefer’s fixed point theorem applied to a sequence of truncated problems. We state a simple uniqueness result. We also generalize the results to Robin boundary conditions. Refs 17.

Keywords:

elliptic, semilinear, locally convex, fixed point, arbitrary domain