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: $$-{\epsilon }^2\Delta v+V(x)v=\frac{1}{{\epsilon }^{\alpha }}(I_{\alpha }\ast F(v))f(v)in{\mathbb{R}}^N,$$ where $N\ge 3$, $\alpha \in (0,N)$, $I_{\alpha }(x)=\frac{A_{\alpha }}{|x{|}^{N-\alpha }}$ 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 $cupl(K)+1$ 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 $K=\{x\in \Omega ;V(x)=m_0\}$.

52 pages

References in corpus (1)

• Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation