# A simple variational approach to weakly coupled competitive elliptic systems

## TL;DR

This paper introduces a straightforward variational framework for identifying fully nontrivial solutions to weakly coupled elliptic systems, linking solutions to critical points of a smooth functional on a product of spheres.

## Contribution

It presents a novel, simple variational approach that simplifies finding solutions to weakly coupled elliptic systems and extends existing results in the field.

## Key findings

- Established a variational setting for the system
- Linked solutions to critical points of a smooth functional
- Extended known results for weakly coupled systems

## Abstract

The main purpose of this paper is to exhibit a simple variational setting for finding fully nontrivial solutions to the weakly coupled elliptic system (1.1). We show that such solutions correspond to critical points of a $\mathcal{C}^1$-functional $\Psi:\mathcal{U}\to\mathbb{R}$ defined in an open subset $\mathcal{U}$ of the product $\mathcal{T}:=S_1\times\cdots\times S_M$ of unit spheres $S_i$ in an appropriate Sobolev space. We use our abstract setting to extend and complement some known results for the system (1.1).