Normalized solutions to nonlinear Schrödinger equations with competing Hartree-type nonlinearities

Abstract

In this paper, we consider solutions to the following nonlinear Schrödinger equation with competing Hartree-type nonlinearities, $$ -Δu + λu=\left(|x|^{-γ_1} \ast |u|^2\right) u - \left(|x|^{-γ_2} \ast |u|^2\right) u\quad \mbox{in} \,\, \R^N, $$ under the $L^2$-norm constraint $${\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=c>0,$$ where $N\ge 1$, $0<{\hat{I}}_2^3<{\hat{I}}_1^3<\min \{N,4\}$ and $\hat{I}»\in \mathbb{R}$ appearing as Lagrange multiplier is unknown. First we establish the existence of ground states in the mass subcritical, critical and supercritical cases. Then we consider the well-posedness and dynamical behaviors of solutions to the Cauchy problem for the associated time-dependent equations.

31 pages