Ground states for a coupled nonlinear Schr\"odinger system

TL;DR

This paper investigates the existence of ground state solutions for a coupled nonlinear Schrödinger system, revealing new existence results across various dimensions and parameter ranges, especially for certain nonlinear exponents.

Contribution

It extends known results by establishing the existence of positive ground states for a broader range of parameters, notably for all dimensions when 1<q<2 and improving results for q>2 in one dimension.

Findings

Existence of ground states for 1<q<2 in all dimensions and for all meters.

Improved existence results for q>2 in one-dimensional case.

Enhanced understanding of parameter regimes for ground state solutions.

Abstract

We study the existence of ground states for the coupled Schr\"odinger system \begin{equation} \label{ellipticabstract} \left\{ \begin{array}{llll} -\Delta u+u&=&|u|^{2q-2}u+b|v|^q|u|^{q-2}u\\ -\Delta v+\omega^2v&=&|v|^{2q-2}v+b|u|^q|v|^{q-2}v \end{array}\right. \end{equation} in $\mathbf{R}^n$, for $\omega \geq 1$, $b>0$ (the so-called "attractive case") and $q>1$ ($q<\frac n{n-2}$ if $n\geq 3$). We improve for several ranges of $(q,n,\omega)$ the known results concerning the existence of positive ground state solutions with non-trivial components. In particular, we prove that for $1<q<2$ such ground states exist in all dimensions and for all values of $\omega$, which constitutes a drastic change of behaviour with respect to the case $q\geq 2$. Furthermore, in the one-dimensional case $n=1$, we improve the results present in the literature for $q>2$.