Ground States for a nonlinear Schr\"odinger system with sublinear coupling terms

TL;DR

This paper proves the existence of radially decreasing ground states for a coupled nonlinear Schrödinger system with sublinear coupling terms, showing positivity in certain parameter regimes across all dimensions.

Contribution

It establishes the existence of nonnegative, radially decreasing ground states for the system with sublinear coupling, including positivity results for small q values.

Findings

Existence of radially decreasing ground states proven.

Ground states are positive for 1<q<2 in all dimensions.

Results hold for symmetric attractive coupling with positive parameters.

Abstract

We study the existence of ground states for the coupled Schr\"odinger system \begin{equation} \left\{\begin{array}{lll} \displaystyle -\Delta u_i+\lambda_i u_i= \mu_i |u_i|^{2q-2}u_i+\sum_{j\neq i}b_{ij} |u_j|^q|u_i|^{q-2}u_i \\ u_i\in H^1(\mathbb{R}^n), \quad i=1,\ldots, d, \end{array}\right. \end{equation} $n\geq 1$, for $\lambda_i,\mu_i >0$, $b_{ij}=b_{ji}>0$ (the so-called "symmetric attractive case") and $1<q<n/(n-2)^+$. We prove the existence of a nonnegative ground state $(u_1^*,\ldots,u_d^*)$ with $u_i^*$ radially decreasing. Moreover we show that, for $1<q<2$, such ground states are positive in all dimensions and for all values of the parameters.