Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data

Abstract

Let $s\in (0,1),$ $1<p<\frac{N}{s}$ and $\Omega \subset {\mathbb{R}}^N$ be an open bounded set. In this work we study the existence of solutions to problems ($E_{\pm }$) $Lu\pm g(u)=\mu$ and $u=0$ a.e. in ${\mathbb{R}}^N\setminus \Omega ,$ where $g\in C(\mathbb{R})$ is a nondecreasing function, $\mu$ is a bounded Radon measure on $\Omega$ and $L$ is an integro-differential operator with order of differentiability $s\in (0,1)$ and summability $p\in (1,\frac{N}{s}).$ More precisely, $L$ is a fractional $p-$Laplace type operator. We establish sufficient conditions for the solvability of problems ($E_{\pm }$). In the particular case $g(t)=|t{|}^{\kappa -1}t;$ $\kappa >p-1,$ these conditions are expressed in terms of Bessel capacities.