On Semilinear Elliptic Equation with Measurable Nonlinearity

TL;DR

This paper investigates a semilinear elliptic equation with a discontinuous, monotone nonlinearity that is not a Carathéodory function, establishing an existence theorem for solutions in bounded domains.

Contribution

It introduces an existence result for elliptic equations with measurable, discontinuous nonlinearities not satisfying Carathéodory conditions.

Findings

Existence of solutions proven for the considered class of equations.

Handling of nonlinearity that is discontinuous and not Carathéodory.

Extension of elliptic theory to more general nonlinearities.

Abstract

We consider a semilinear elliptic equation in a bounded domain with zero boundary conditions. The nonlinearity is discontinuous and monotone, but it is not a Carath\'eodory's function. The existence theorem has been proved.