Particular existence of positive solutions for elliptic equations involving regional fractional Laplacian of order $(0,\frac12]$

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.