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

In this paper, using variational methods, we study multiplicity of multi-bump solutions for the following nonlinear magnetic Choquard equation {-(del+iA(x))(2)u+(lambda V(x) +1)u =(1/vertical bar x vertical bar(mu) * vertical bar u vertical bar(p))vertical bar u vertical bar(p-2)u x is an element of R-N, u is an element of H-1(R-N, C) where N >= 2, lambda > 0 is a real parameter, 0 < mu < 2, i is the imaginary unit, p is an element of (2, 2*(2(N-mu)/2N)), where 2* = 2N/N-2 if N >= 3, 2* +infinity, if N = 2. The magnetic potential A is an element of L-loc(2) (R-N , R-N) and V : R-N -> R is a nonnegative continuous function. We show that if the zero set of V has several isolated connected components Omega(1), ..., Omega(k) such that the interior of Omega(j) is non-empty and partial derivative Omega(j) is smooth, then for h > 0 large enough, the above equation has at least 2(k) - 1 multi-bump solutions. (C) 2021 Elsevier Inc. All rights reserved.