Boundary value problem with measures for fractional elliptic equations involving source nonlinearities

TL;DR

This paper studies positive solutions of fractional elliptic equations with measure boundary conditions, establishing a priori estimates, existence results, and critical exponents for nonlinearities like power functions.

Contribution

It introduces a universal a priori estimate for solutions, proves existence with measure boundary data, and identifies a critical exponent and solution thresholds for power nonlinearities.

Findings

Established a priori bounds for solutions and their gradients.

Proved existence of solutions with measure boundary conditions.

Identified a critical exponent and solution thresholds for power nonlinearities.

Abstract

We are concerned with positive solutions of equation (E) $(-\Delta)^s u=f(u)$ in a domain $\Omega \subset \mathbb{R}^N$ ($N>2s$), where $s \in (\frac{1}{2},1)$ and $f\in C^{\alpha}_{loc}(\mathbb{R})$ for some $\alpha \in(0,1)$. We establish a universal a priori estimate for positive solutions of (E), as well as for their gradients. Then for $C^2$ bounded domain $\Omega$, we prove the existence of positive solutions of (E) with prescribed boundary value $\rho \nu$, where $\rho>0$ and $\nu$ is a positive Radon measure on $\partial \Omega$ with total mass $1$, and discuss regularity property of the solutions. When $f(u)=u^p$, we demonstrate that there exists a critical exponent $p_s:=\frac{N+s}{N-s}$ in the following sense. If $p\geq p_s$, the problem does not admit any positive solution with $\nu$ being a Dirac mass. If $p\in(1,p_s)$ there exits a threshold value $\rho^*>0$ such that for $\rho\in (0, \rho^*]$, the problem admits a positive solution and for $\rho>\rho^*$, no positive solution exists. We also show that, for $\rho>0$ small enough, the problem admits at least two positive solutions.