Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data

TL;DR

This paper investigates the existence of solutions to nonlinear nonlocal fractional p-Laplace equations with measure data, establishing conditions based on Bessel capacities for solvability in bounded domains.

Contribution

It provides new sufficient conditions for the existence of solutions to fractional nonlocal equations with measure data, especially for power nonlinearities, using capacity theory.

Findings

Established solvability conditions involving Bessel capacities.

Extended existence results to nonlinearities of power type.

Analyzed equations with measure data and nonlocal operators.

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.