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, Iα 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.

中文翻译：

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半经典 Choquard 方程的更高拓扑类型的多重束缚态

我们关注 Choquard 方程的半经典状态$$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v \in H^1({\mathbb R}^N),$$在哪里ñ⩾ 2,一世α是阶数为 α ∈ (0,ñ− 1) 和 2 ⩽p< (ñ+ α)/(ñ- 2)。当潜力五假设有界且远离零，我们构建了一个更高拓扑类型的局部束缚态族，它们集中在势能的局部最小点周围五ε → 0。这些解是通过结合 Byeon-Wang 的惩罚方法和经典的对称山口定理获得的。