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

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