Uniqueness of positive bound states to Schrodinger systems with critical exponents

TL;DR

This paper proves the uniqueness and radial symmetry of positive solutions for a class of elliptic systems with critical exponents, relevant to Bose-Einstein condensates, extending understanding of such nonlinear PDEs.

Contribution

It establishes the uniqueness and radial symmetry of positive solutions for elliptic systems with critical exponents, including cases related to Bose-Einstein condensates.

Findings

Positive solutions are unique for the given elliptic systems.

Positive solutions exhibit radial symmetry.

Results apply to systems with critical exponents in higher dimensions.

Abstract

We prove the uniqueness for the positive solutions of the following elliptic systems: \begin{eqnarray*} \left\{\begin{array}{ll} - \lap (u(x)) = u(x)^{\alpha}v(x)^{\beta} - \lap (v(x)) = u(x)^{\beta} v(x)^{\alpha} \end{array} \right. \end{eqnarray*} Here $x\in R^n$, $n\geq 3$, and $1\leq \alpha, \beta\leq \frac{n+2}{n-2}$ with $\alpha+\beta=\frac{n+2}{n-2}$. In the special case when $n=3$ and $\alpha =2, \beta=3$, the systems come from the stationary Schrodinger system with critical exponents for Bose-Einstein condensate. As a key step, we prove the radial symmetry of the positive solutions to the elliptic system above with critical exponents.