# 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.