Existence and symmetry of positive ground states for a doubly critical Schrodinger system

TL;DR

This paper investigates the existence and symmetry of positive ground state solutions for a doubly critical Schrödinger system with coupling, highlighting how the least energy depends on parameters and employing various methods for different cases.

Contribution

It establishes the existence of positive ground states for a doubly critical Schrödinger system using novel methods tailored to parameter ranges, including a variational perturbation approach for small coupling.

Findings

Positive ground state solutions exist for various parameter ranges.

The least energy level depends on the relations among b1, b1, and 2.

Solutions are radially symmetric and obtained via different methods.

Abstract

We study the following doubly critical Schr\"{o}dinger system $$-\Delta u -\frac{\la_1}{|x|^2}u=u^{2^\ast-1}+ \nu \al u^{\al-1}v^\bb, \quad x\in \RN, -\Delta v -\frac{\la_2}{|x|^2}v=v^{2^\ast-1} + \nu \bb u^{\al}v^{\bb-1}, \quad x\in \RN, u, v\in D^{1, 2}(\RN),\quad u, v>0 in $\RN\setminus{0}$},$$ where $N\ge 3$, $\la_1, \la_2\in (0, \frac{(N-2)^2}{4})$, $2^\ast=\frac{2N}{N-2}$ and $\al>1, \bb>1$ satisfying $\al+\bb=2^\ast$. This problem is related to coupled nonlinear Schr\"{o}dinger equations with critical exponent for Bose-Einstein condensate. For different ranges of $N$, $\al$, $\bb$ and $\nu>0$, we obtain positive ground state solutions via some quite different methods, which are all radially symmetric. It turns out that the least energy level depends heavily on the relations among $\al, \bb$ and 2. Besides, for sufficiently small $\nu>0$, positive solutions are also obtained via a variational perturbation approach. Note that the Palais-Smale condition can not hold for any positive energy level, which makes the study via variational methods rather complicated.