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Our purpose of this paper is to investigate positive solutions of theelliptic 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 regionalfractional Laplacian and $h$ is the nonlinearity.  Ordinarily, positive solutions vanishing at the boundary are not anticipatedto be derived for the equations with regional fractional Laplacian of order$s\in(0,\frac12]$. Positive solutions are obtained when the nonlinearityassumes 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, radiallysymmetric and decreasing functions under suitable conditions.

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