A study of second order semilinear elliptic PDE involving measures

TL;DR

This paper investigates the existence and uniqueness of very weak solutions for second order semilinear elliptic PDEs involving measures and L^1 functions, highlighting conditions under which solutions exist or fail to exist.

Contribution

It introduces the concept of reduced limits for measures and establishes criteria for the existence of very weak solutions in this context.

Findings

Existence of very weak solutions for L^1 functions.

Non-existence of solutions when replacing L^1 functions with measures.

Introduction of reduced limits to characterize solvability.

Abstract

The objective of this article is to study the boundary value problem for the general semilinear elliptic equation of second order involving $L^1$ functions or Radon measures with finite total variation. The study investigates the existence and uniqueness of `{\it very weak}' solutions to the boundary value problem for a given $L^1$ function. However, a `{\it very weak}' solution need not exist when an $L^1$ function is replaced with a measure due to which the corresponding reduced limits has been found for which the problem admits a solution in a `{\it very weak}' sense.