Compactness of Green operators with applications to semilinear nonlocal elliptic equations

TL;DR

This paper investigates the compactness properties of Green operators associated with integro-differential operators on bounded domains and applies these results to establish existence conditions for solutions to semilinear nonlocal elliptic equations involving Radon measures.

Contribution

It introduces unified techniques for analyzing fractional operators and derives sharp compactness and existence results in weighted spaces, extending prior research.

Findings

Established sharp compactness of Green operators in weighted spaces

Proved existence of solutions under subcriticality and capacity conditions

Extended results to various fractional and nonlocal operators

Abstract

In this paper, we consider a class of integro-differential operators $\mathbb{L}$ posed on a $C^2$ bounded domain $\Omega \subset \mathbb{R}^N$ with appropriate homogeneous Dirichlet conditions where each of which admits an inverse operator commonly known as the Green operator $\mathbb{G}^{\Omega}$. Under mild conditions on $\mathbb{L}$ and its Green operator, we establish various sharp compactness of $\mathbb{G}^{\Omega}$ involving weighted Lebesgue spaces and weighted measure spaces. These results are then employed to prove the solvability for semilinear elliptic equation $\mathbb{L} u + g(u) = \mu$ in $\Omega$ with boundary condition $u=0$ on $\partial \Omega$ or exterior condition $u=0$ in $\mathbb{R}^N \setminus \Omega$ if applicable, where $\mu$ is a Radon measure on $\Omega$ and $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing continuous function satisfying a subcriticality integral condition. When $g(t)=|t|^{p-1}t$ with $p>1$, we provide a sharp sufficient condition expressed in terms of suitable Bessel capacities for the existence of a solution. The contribution of the paper consists of (i) developing novel unified techniques which allow to treat various types of fractional operators and (ii) obtaining sharp compactness and existence results in weighted spaces, which refine and extend several related results in the literature.