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Our purpose in this article to show the existence of positive classicalsolutions of $$  ( - \Delta )_{B_1}^s u +u=h_1 u^p+\epsilon h_2 \quad  {\rm in} \ \, B_1,\qquad u = 0 \quad {\rm on}\ \partial B_1  $$ for $\epsilon>0$ small enough, where $( - \Delta )_{B_1}^s$ is theregional fractional Laplacian, $p>1$, $h_i$ with $i=1,2$ are H\"oldercontinuous and satisfy some additional conditions. Our existence is based onthe solution of $$  ( - \Delta )_{B_1}^s u +u=1 \quad  {\rm in} \ \, B_1,\qquad u = 0 \quad {\rm on}\ \partial B_1.  $$

On Schr\"odinger equations involving the regional fractional Laplacian in a ball with the zero boundary condition

On Schr\"odinger equations involving the regional fractional Laplacian in a ball with the zero boundary condition