Multi-Bump Solutions for Nonlinear Choquard Equation with Potential Wells and a General Nonlinearity

In this article, we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity −Δu+(λa(x)+1)u=(1|x|α*F(u))f(u)inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta u + (\lambda a(x) + 1)u = \left(\frac{1}{|x|^\alpha}* F(u)\right) f(u) \; in \; \mathbb{R}^N,$$\end{document} where N ≥ 3, 0 < α < min{N, 4}, λ is a positive parameter and the nonnegative potential function a(x) is continuous. Using variational methods, we prove that if the potential well int(a−1(0)) consists of k disjoint components, then there exist at least 2k − 1 multi-bump solutions. The asymptotic behavior of these solutions is also analyzed as λ → +∞.