Minimal energy solutions for repulsive nonlinear Schr\"odinger systems

Abstract

In this paper we establish existence and nonexistence results concerning fully nontrivial minimal energy solutions of the nonlinear Schr\"odinger system \begin{align*} \begin{gathered} -\Delta u + \, u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q \quad\text{in}\R^n, -\Delta v + \omega^2 v = |v|^{2q-2}v + b|u|^q|v|^{q-2}v\quad\text{in}\R^n. \end{gathered} \end{align*} We consider the repulsive case $b<0$ and assume that the exponent $q$ satisfies $1<q<\frac{n}{n-2}$ in case $n\geq 3$ and $1<q<\infty$ in case $n=1$ or $n=2$. For space dimensions $n\geq 2$ and arbitrary $b<0$ we prove the existence of fully nontrivial nonnegative solutions which converge to a solution of some optimal partition problem as $b\to -\infty$. In case $n=1$ we prove that minimal energy solutions exist provided the coupling parameter $b$ has small absolute value whereas fully nontrivial solutions do not exist if $1<q\leq 2$ and $b$ has large absolute value.