Existence of solutions to elliptic equations involving regional fractional Laplacian with order $(0,\frac12]$

TL;DR

This paper establishes the existence of positive solutions for elliptic equations involving the regional fractional Laplacian of order up to 1/2, under specific nonlinearities and boundary conditions.

Contribution

It demonstrates the existence of positive solutions for elliptic equations with regional fractional Laplacian of order s in (0, 1/2], a case previously not well-understood.

Findings

Positive solutions exist for certain nonlinearities.

Solutions are obtained under conditions on the nonlinearity functions.

The results extend the understanding of fractional Laplacian equations with boundary conditions.

Abstract

Our purpose of this paper is to investigate positive solutions of the elliptic equation with regional fractional Laplacian $$ ( - \Delta )_{B_1}^s u +u= h(x,u) \quad {\rm in} \ \, B_1,\qquad u\in C_0(B_1), $$ where $( - \Delta )_{B_1}^s$ with $s\in(0,\frac12]$ is the regional fractional Laplacian and $h$ is the nonlinearity. Ordinarily, positive solutions vanishing at the boundary are not anticipated to be derived for the equations with regional fractional Laplacian of order $s\in(0,\frac12]$. Positive solutions are obtained when the nonlinearity assumes the following two models: $h(x,t)=f(x)$ or $h(x,t)=h_1(x)\, t^p+ \epsilon h_2(x)$, where $p>1$, $\epsilon>0$ small and $f, h_1, h_2$ are H\"older continuous, radially symmetric and decreasing functions under suitable conditions.