Existence of solution for a class of elliptic problems in exterior domain with discontinuous nonlinearity

Abstract

In this paper, we study the existence of nontrivial solution for a class of elliptic problems of the form $$\begin{aligned} -\Delta u+u = f_{p, \delta }(u(x)) \quad {\text {a.e \, in}} \quad \Omega \end{aligned}$$ where $$\Omega \subset \mathbb {R}^N$$ is an exterior domain for $$N>2$$ and $$f_{p, \delta }:\mathbb {R} \rightarrow \mathbb {R}$$ is an odd discontinuous function given by $$\begin{aligned} f_{p, \delta }(t) = {\left\{ \begin{array}{ll} t|t|^{p-2}, &{} t \in [0, a],\\ (1 + \delta )t|t|^{p-2}, &{} t > a, \end{array}\right. } \end{aligned}$$ with $$ a>0,\; \delta > 0$$ and $$p \in (2, 2^*)$$ . For small enough $$\delta $$ and a, seeking help of the dual functional corresponding to the problem, we prove existence of at least one positive solution when $$\mathbb R^N {\setminus } \Omega \subset B_{\sigma }(0)$$ for sufficiently small $$\sigma $$ .