Semilinear fractional elliptic equations with gradient nonlinearity involving measures

TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of solutions to a fractional elliptic equation with gradient nonlinearity involving measures, expanding understanding of such equations in bounded domains.

Contribution

It establishes the existence of weak solutions for subcritical nonlinearities and describes their asymptotic behavior and uniqueness in specific cases.

Findings

Existence of weak solutions for subcritical g

Asymptotic behavior characterized for Dirac measure data

Uniqueness established when g(s)=s^p, p≥1, and ε=1

Abstract

We study the existence of solutions to the fractional elliptic equation (E1) $(-\Delta)^\alpha u+\epsilon g(|\nabla u|)=\nu $ in a bounded regular domain $\Omega$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $\Omega^c$, where $\epsilon=1$ or $-1$, $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(1/2,1)$, $\nu$ is a Radon measure and $g:\R_+\mapsto\R_+$ is a continuous function. We prove the existence of weak solutions for problem (E1)-(E2) when $g$ is subcritical. Furthermore, the asymptotic behavior and uniqueness of solutions are described when $\nu$ is Dirac mass, $g(s)=s^p$, $p\geq 1$ and $\epsilon=1$.