Minimal energy solutions for repulsive nonlinear Schr\"odinger systems

TL;DR

This paper investigates the existence and nonexistence of minimal energy solutions for a nonlinear Schrödinger system with repulsive interactions, revealing conditions under which solutions exist or do not, depending on parameters and space dimensions.

Contribution

It establishes new existence results for fully nontrivial solutions in higher dimensions and characterizes nonexistence in one dimension for certain parameter ranges.

Findings

Existence of solutions for $n extgreater=2$ and arbitrary negative $b$.

Solutions converge to an optimal partition problem as $b o - abla$.

Nonexistence of solutions in 1D for large $|b|$ when $1<q extless=2$.

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.