Multi-bump solutions for the nonlinear magnetic Choquard equation with deepening potential well

Abstract

In this paper, using variational methods, we study multiplicity of multi-bump solutions for the following nonlinear magnetic Choquard equation{−(∇+iA(x))2u+(λV(x)+1)u=(1|x|μ⁎|u|p)|u|p−2ux∈RN,u∈H1(RN,C), where N≥2, λ>0 is a real parameter, 0<μ<2, i is the imaginary unit, p∈(2,2⁎(2(N−μ)2N)), where 2⁎=2NN−2 if N≥3, 2⁎=+∞, if N=2. The magnetic potential A∈Lloc2(RN,RN) and V:RN→R is a nonnegative continuous function. We show that if the zero set of V has several isolated connected components Ω1,⋯,Ωk such that the interior of Ωj is non-empty and ∂Ωj is smooth, then for λ>0 large enough, the above equation has at least 2k−1 multi-bump solutions.