Revista Matemática Iberoamericana


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Volume 35, Issue 6, 2019, pp. 1885–1924
DOI: 10.4171/rmi/1105

Published online: 2019-09-02

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

Silvia Cingolani[1] and Kazunaga Tanaka[2]

(1) Università degli Studi di Bari Aldo Moro, Italy
(2) Waseda University, Tokyo, Japan

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation $$ -\varepsilon^2\Delta v+V(x) v = \frac{1}{\varepsilon^\alpha}\,(I_\alpha*F(v))f(v) \quad \hbox{in } \mathbb{R}^N,$$ 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 $K=\{x\in\Omega;$ $\, V(x)=m_0\}$.

Keywords: Nonlinear Choquard equation, semiclassical states, non-local nonlinearities, positive solutions, potential well, relative cup-length

Cingolani Silvia, Tanaka Kazunaga: Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well. Rev. Mat. Iberoam. 35 (2019), 1885-1924. doi: 10.4171/rmi/1105