On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields

Abstract

In this paper, we are concerned with the existence of least energy solutions of nonlinear Schrödinger equations with electromagnetic fields − ( ∇ + i A ( x ) ) 2 u ( x ) + ( λ a ( x ) + 1 ) u ( x ) = | u | p − 2 u , x ∈ R N for sufficiently large λ , where i is the imaginary unit, 2 < p < 2 N N − 2 for N ≥ 3 and 2 < p < + ∞ for N = 1 , 2 . a ( x ) is a real continuous function on R N , and A ( x ) = ( A 1 ( x ) , A 2 ( x ) , … , A N ( x ) ) is such that A j ( x ) is a real local Hölder continuous function on R N for j = 1 , 2 , … , N . Using variational methods we prove the existence of least energy solution u ( x ) which localizes near the potential well int ( a − 1 ( 0 ) ) for λ large.