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 $0<\alpha <N$, $\frac{N+\alpha }{N}<p< \frac{N+\alpha }{N-2}$ if $N\ge 3$, $\frac{N+\alpha }{N}<p<\infty$ if $N=1,2$ and $I_\alpha$ is the Riesz potential. The external potential $V: {\mathbb {R}}^N \rightarrow {\mathbb {R}}$ is assumed to satisfy (A1) $V \in L^q({\mathbb {R}}^N) + L^\infty ({\mathbb {R}}^N)$ with $q=1$ if $N=1$, $q>1$ 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.