Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in R2

Abstract

In this paper, using variational methods, we establish the existence and multiplicity of multi-bump solutions for the following nonlinear magnetic Schrödinger equation -(∇+iA(x))2u+(λV(x)+Z(x))u=f(|u|2)uinR2,where λ> 0 , f(t) is a continuous function with exponential critical growth, the magnetic potential A: R2→ R2 is in Lloc2(R2) and the potentials V, Z: R2→ R are continuous functions verifying some natural conditions. We show that if the zero set of the potential V has several isolated connected components Ω 1, … , Ω k such that the interior of Ω j is non-empty and ∂Ω j is smooth, then for λ> 0 large enough, the equation has at least 2 k- 1 multi-bump solutions.