An Existence Result for a Class of Magnetic Problems in Exterior Domains

Abstract

AbstractIn this paper we deal with the existence of solutions for the following class of magnetic semilinear Schrödinger equation $$\begin{aligned} (P) \qquad \qquad \left\{ \begin{aligned}&(-i\nabla + A(x))^2u +u = |u|^{p-2}u,\;\;\text{ in }\;\;\Omega ,\\&u=0\;\;\text{ on }\;\;\partial \Omega , \end{aligned} \right. \end{aligned}$$ ( P ) ( - i ∇ + A ( x ) ) 2 u + u = | u | p - 2 u , in Ω , u = 0 on ∂ Ω , where $$N \ge 3$$ N ≥ 3 , $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is an exterior domain, $$p\in (2, 2^*)$$ p ∈ ( 2 , 2 ∗ ) with $$2^*=\frac{2N}{N-2}$$ 2 ∗ = 2 N N - 2 , and $$A: {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$$ A : R N → R N is a continuous vector potential verifying $$A(x) \rightarrow 0\;\;\text{ as }\;\;|x|\rightarrow \infty .$$ A ( x ) → 0 as | x | → ∞ .