Uniqueness results for semilinear elliptic systems on $\R^n$

TL;DR

This paper establishes criteria for the uniqueness of positive radially symmetric solutions to semilinear elliptic systems on ^n, with applications to nonlinear Schrf6dinger systems, providing new conditions for the existence of unique solutions.

Contribution

It introduces new sufficient and necessary conditions for the uniqueness of positive solutions in semilinear elliptic systems, extending previous results to more general cases.

Findings

Uniqueness criteria depend on parameters b, q, n.

Results are optimal for the case n=1.

Generalization of Wei and Yao's results for q=2.

Abstract

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form \begin{align*} \begin{aligned} - \Delta u &= f(|x|,u,v)\quad\text{in}\R^n, - \Delta v &= f(|x|,v,u)\quad\text{in}\R^n. \end{aligned} \end{align*} As an application we consider the following nonlinear Schr\"odinger system \begin{align*} \begin{aligned} - \Delta u + u &= u^{2q-1} + b u^{q-1}v^q\quad\text{in}\R^n, - \Delta v + v &= v^{2q-1} + b v^{q-1}u^q \quad\text{in}\R^n. \end{aligned} \end{align*} for $b>0$ and exponents $q$ which satisfy $1<q<\infty$ in case $n\in\{1,2\}$ and $1<q<\frac{n}{n-2}$ in case $n\geq 3$. Generalizing the results of Wei and Yao dealing with the case $q=2$ we find new sufficient conditions and necessary conditions on $b,q,n$ such that precisely one positive solution exists. Our results dealing with the special case $n=1$ are optimal.