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

Abstract

AbstractWe are concerned with the semi-classical states for the Choquard equation$$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$whereN⩾ 2,Iαis the Riesz potential with order α ∈ (0,N− 1) and 2 ⩽p< (N+ α)/(N− 2). When the potentialVis 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 potentialVas ε → 0. These solutions are obtained by combining the Byeon–Wang's penalization approach and the classical symmetric mountain pass theorem.