Ground states for a coupled nonlinear Schr\"odinger system

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 $\omega \ge 1$, $b>0$ (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 $q>2$.