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Multiple bound states of higher topological type for semi-classical Choquard equations. (English) Zbl 1459.35178

Summary: 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 \(N\geq 2\), \(I_\alpha\) is the Riesz potential with order \(\alpha\in (0,N-1)\) and \(2\leq p<(N+\alpha)/(N-2)\). 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 \(\epsilon\rightarrow 0\). These solutions are obtained by combining the Byeon-Wang’s penalization approach and the classical symmetric mountain pass theorem.

MSC:

35J61 Semilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J20 Variational methods for second-order elliptic equations
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