We consider the following fractional elliptic problem:
$$\begin{aligned} (P)\left\{ \begin{array}{ll} (-\Delta )^s u = f(u) H(u-\mu )&{} \quad \text{ in } \ \Omega ,\\ u =0 &{}\quad \text{ on } \ \mathbb{{R}}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$
where
\((-\Delta )^s, s\in (0,1)\) is the fractional Laplacian,
\(\Omega \) is a bounded domain of
\(\mathbb{{R}}^n,(n\ge 2s)\) with smooth boundary
\(\partial \Omega ,\) H is the Heaviside step function,
f is a given function and
\(\mu \) is a positive real parameter. The problem (
P) can be considered as simplified version of some models arising in different contexts. We employ variational techniques to study the existence and multiplicity of positive solutions of problem (
P).