Blow-up behavior of prescribed mass minimizers for nonlinear Choquard equations with singular potentials

Abstract

We study the constrained minimizing problem for the energy functional related to the nonlinear Choquard equation $$\begin{aligned} I(a) = \inf \left\{ E(\phi ) \ : \ \phi \in H^1\big ({\mathbb {R}}^N\big ), \Vert \phi \Vert ^2_{L^2} =a \right\} , \end{aligned}$$where $$N\ge 1$$, $$a>0$$, $$\begin{aligned} E(\phi ) := \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla \phi (x)|^2 dx + \frac{1}{2} \int _{{\mathbb {R}}^N} V(x) |\phi (x)|^2 dx -\frac{1}{2p} \int _{{\mathbb {R}}^N} (I_\alpha * |\phi |^p)(x) |\phi (x)|^p dx \end{aligned}$$is the energy functional with $$0 1$$ if $$N=2$$ and $$q=\frac{N}{2}$$ if $$N\ge 3$$ and (A2) for any $$c>0$$, $$|\{x \in {\mathbb {R}}^N \ : \ |V(x)| >c \}| <\infty $$. We first give a complete classification of existence and non-existence of minimizers for the problem. In the mass-critical case $$p=\frac{N+\alpha +2}{N}$$, under an appropriate assumption of the external potential, we give a detailed description of the blow-up behavior of minimizers as the mass tends to a critical value.