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