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

Abstract

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

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

We develop a new variational approach and we show the existence of a family of solutions concentrating, as $\epsilon \to 0$, 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 $p\in (\frac{N+\alpha }{N},\frac{N+\alpha }{N-2})$. Especially, we can give the existence result for locally sublinear case $p\in (\frac{N+\alpha }{N},2)$, 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 $\mathrm{cupl}(K)+1$ solutions concentrating around $K$ as $\epsilon \to 0$, 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 $K=\{x\in \Omega ;$ $V(x)=m_0\}$.