Nonlinear elliptic equations with integro-differential divergence form operators and measure data under sign condition on the nonlinearity

TL;DR

This paper investigates the existence of solutions for nonlinear elliptic equations involving integro-differential operators and measure data, under a sign condition, using sub and supersolution methods.

Contribution

It introduces a new approach to establish the existence of solutions for nonlinear elliptic equations with measure data and integro-differential operators under minimal assumptions.

Findings

Existence of maximal measure for which solutions exist

Application of sub and supersolution method to integro-differential operators

Introduction of reduced measure concept in this context

Abstract

We study existence problem for semilinear equations with Borel measure data and operator generated by a symmetric Markov semigroup. We assume merely that the nonlinear part satisfies the so-called sign condition. Using the method of sub and supersolutions we show the existence of maximal measure for which there exists a solution to the problem (the so-called reduced measure introduced by H. Brezis, M. Marcus and A.C. Ponce).