# Semi-linear cooperative elliptic systems involving Schr{\"o}dinger operators: Groundstate positivity or negativity

## TL;DR

This paper investigates the positivity or negativity of ground state solutions to a semi-linear cooperative Schrödinger system with a potential that grows at infinity, analyzing how solutions behave near the principal eigenvalue.

## Contribution

It provides a detailed analysis of the sign of ground state solutions for a class of semi-linear cooperative Schrödinger systems involving variable potentials and eigenvalue parameters.

## Key findings

- Ground state solutions can be positive or negative depending on parameters.
- The behavior of solutions is characterized near the principal eigenvalue.
- The variational approach is used to analyze the system.

## Abstract

We study here the behavior of the solutions to a $2\times 2$ semi-linear cooperative system involving Schr\" odinger operators (considered in its variational form): $$LU:=(-\Delta + q(x))U = AU+\mu U + F(x,U) \quad{\rm in}\ \mathbb{R}^N$$ $$U(x)_{|x|\rightarrow \infty} \rightarrow 0$$ where $q$ is a continuous positive potential tending to $+\infty$ at infinity; $\mu$ is a real parameter varying near the principal eigenvalue of the system; $U$ is a column vector with components $u_1$ and $u_2$ and $A$ is a square cooperative matrix with constant coefficient. $F$ is a column vector with components $f_1$ and $f_2$ depending eventually on $U$.