Existence of solutions to elliptic equations involving regional fractional Laplacian with order $(0,\frac{1}{2}]$

Abstract

Our purpose of this paper is to investigate positive solutions of the elliptic equation with regional fractional Laplacian $$(-\Delta {)}_{B_1}^{s}u+u=h(x,u)\mathrm{in}{B}_1,u\in C_0(B_1),$$ where $(-\Delta {)}_{B_1}^s$ with $s\in (0,\frac{1}{2}]$ 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 $s\in (0,\frac{1}{2}]$. 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 Hölder continuous, radially symmetric and decreasing functions under suitable conditions.

30