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We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation
where , , is the Riesz potential, , and is a small parameter.
We develop a new variational approach and we show the existence of a family of solutions concentrating, as , to a local minima of under general conditions on . Our result is new also for and applicable for . Especially, we can give the existence result for locally sublinear case , which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen.
We also study the multiplicity of positive single-peak solutions and we show the existence of at least solutions concentrating around~ as , where is the set of minima of in a bounded potential well , that is, and .
Cite this article
Silvia Cingolani, Kazunaga Tanaka, Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well. Rev. Mat. Iberoam. 35 (2019), no. 6, pp. 1885–1924DOI 10.4171/RMI/1105