Compactness of Green operators with applications to semilinear nonlocal elliptic equations

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.