Normalized ground states for the lower critical fractional Choquard equation with a focusing local perturbation

Abstract

In this paper, we study the existence of normalized ground states to the following lower critical fractional Choquard equation $ (-\Delta)^su = \lambda u+\gamma(I_{\alpha}*|u|^{1+\frac{\alpha}{N}})|u|^{\frac{\alpha}{N}-1}u+\mu |u|^{q-2}u\ \mbox{in}\ \mathbb R^N $ under the $ L^2 $-norm constraint$ \int_{\mathbb R^N}|u|^2dx = a^2, $where $ N \geq3 $, $ s\in(0,1) $, $ \alpha\in (0,N) $, $ a, \gamma, \mu>0 $ and $ 2<q\leq 2_s^*: = 2N/(N-2s) $. Under suitable restrictions on $ a, \gamma $ and $ \mu $, we prove nonexistence, existence and symmetry of normalized ground states. Specifically, using the extremal function with construction technique, we establish the existence of radially normalized ground states without any restrictions under the $ L^2 $-subcritical perturbation, i.e. $ 2<q<2+4s/N $. In the $ L^2 $-supercritical case $ 2+4s/N<q<2_s^* $, we introduce the homotopy-stable family to establish the existence of Palais-Smale sequence, and the compactness of this sequence to illustrate the existence of normalized ground states. In particular, we consider the fractional Sobolev critical case $ q = 2_s^* $, which corresponds to equations involving double critical terms and is rarely studied in the existing literatures. With the aid of the Sobolev subcritical approximation method, we also obtain the existence of normalized ground states.

Topics & Concepts