Multiple bound states of higher topological type for semi-classical Choquard equations

作者：Xiaonan Liu, Shiwang Ma, Jiankang Xia · 发表于：Proceedings of the Royal Society of Edinburgh Section A Mathematics · 年份：2020 · DOI：10.1017/prm.2020.17 · 被引用次数：8 · 研究领域：Advanced Mathematical Physics Problems、Nonlinear Partial Differential Equations、Black Holes and Theoretical Physics

Abstract We are concerned with the semi-classical states for the Choquard equation $$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$ where N ⩾ 2, I α is the Riesz potential with order α ∈ (0, N − 1) and 2 ⩽ p < ( N + α)/( N − 2). When the potential V is assumed to be bounded and bounded away from zero, we construct a family of localized bound states of higher topological type that concentrate around the local minimum points of the potential V as ε → 0. These solutions are obtained by combining the Byeon–Wang's penalization approach and the classical symmetric mountain pass theorem.