A discontinuous semilinear problem involving the fractional Laplacian

Abstract

We study the following fractional problem of the type$$
(P)\left\{\begin{array}{ll} (-\Delta )^s u = f(u) & \quad \mbox{
in }\ \Omega,
\\[0.3cm]u =0 &\quad \mbox{ on }\
\R^n \setminus \Omega,
\end{array}
\right. $$ where $(-\Delta)^s$ is the fractional Laplacian,
$\Omega$ is a smooth bounded domain in $\R^n,$
$f$ is a discontinuous nonlinearity. Under different growth assumptions on $f$, we obtain various existence results via the nonsmooth minimax theorems.