Ground states for a coupled nonlinear Schrödinger system

Abstract

We study the existence of ground states for the coupled Schrödinger system \begin{equation} \label{ellipticabstract} \left\{ \begin{array}{llll} -Δu+u&=&|u|^{2q-2}u+b|v|^q|u|^{q-2}u\\ -Δv+ω^2v&=&|v|^{2q-2}v+b|u|^q|v|^{q-2}v \end{array}\right. \end{equation} in ${\mathbf{R}}^n$, for $\omega \ge 1$, $b>0$ (the so-called "attractive case") and $q>1$ ($q<\frac{n}{n-2}$ if $n\ge 3$). 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 $q\ge 2$. Furthermore, in the one-dimensional case $n=1$, we improve the results present in the literature for $q>2$.