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Multiple bound states of higher topological type for semi-classical Choquard equations
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 1 / February 2021
- Published online by Cambridge University Press:
- 04 March 2020, pp. 329-355
- Print publication:
- February 2021
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- Article
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We are concerned with the semi-classical states for the Choquard equation
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.
$$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$
