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

TL;DR

This paper proves the existence of weak solutions for semilinear elliptic boundary value problems on any open set, using fixed point methods, with minimal assumptions on the domain and boundary conditions.

Contribution

It establishes existence results without regularity assumptions on the domain and extends to Robin boundary conditions, broadening applicability.

Findings

Existence of weak solutions on arbitrary open sets.

Applicability to Robin boundary conditions.

Use of Schaefer's fixed point theorem for proofs.

Abstract

We show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set. We make no assumptions about the open set, very mild regularity assumptions on the semilinearity, plus a coerciveness assumption which depends on the optimal Poincare-Steklov constant. 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.