# Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well

### Silvia Cingolani

Università degli Studi di Bari Aldo Moro, Italy### Kazunaga Tanaka

Waseda University, Tokyo, Japan

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## Abstract

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation

where $N\geq 3$, $\alpha\in (0,N)$, $I_\alpha(x)={A_\alpha/ |x|^{N-\alpha}}$ is the Riesz potential, $F\in C^1(\mathbb{R},\mathbb{R})$, $F'(s) = f(s)$ and $\varepsilon>0$ is a small parameter.

We develop a new variational approach and we show the existence of a family of solutions concentrating, as $\varepsilon\to 0$, to a local minima of $V(x)$ under general conditions on $F(s)$. Our result is new also for $f(s)=|s|^{p-2}s$ and applicable for $p\in ({N+\alpha\over N}, {N+\alpha\over N-2})$. Especially, we can give the existence result for locally sublinear case $p\in ({N+\alpha\over N},2)$, 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 ${\rm cupl}(K)+1$ solutions concentrating around~$K$ as $\varepsilon\to 0$, where $K\subset \Omega$ is the set of minima of $V(x)$ in a bounded potential well $\Omega$, that is, $m_0 \equiv \inf_{x\in \Omega} V(x) < \inf_{x\in \partial\Omega}V(x)$ and $K=\{x\in\Omega;$ $\, V(x)=m_0\}$.

## 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. 1885–1924

DOI 10.4171/RMI/1105