论文标题
具有竞争性Hartree型非线性的非线性Schrödinger方程的归一化解决方案
Normalized solutions to nonlinear Schrödinger equations with competing Hartree-type nonlinearities
论文作者
论文摘要
在本文中，我们考虑了以下非线性schrödinger方程的解决方案，具有竞争性的hartree-type非线性，$$-ΔU +λu= \ left（| x | x |^{ - γ_1} \ ast | ast | u | u |^2 \ right） \ mbox {in} \，\，\，\ r^n，$$下的$ l^2 $ -norm约束$ \ int \ int \ int \ int \ in {\ r^n} | u | u |^2 \，dx = c> 0，$ n \ geq 1 $，$ n \ geq 1 $，$ 0 <γ_2<γ_2<γ_2<γ_1<gγ_1<q $以拉格朗日乘数出现是未知的。首先，我们在大规模亚临界，批判性和超临界案例中建立了基态的存在。然后，我们考虑了相关时间依赖的方程的解决方案的解决方案的适当性和动力学行为。
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 $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<γ_2 < γ_1 <\min\{N, 4\}$ and $λ\in \R$ 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.
