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 -\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\} .