Normalized solutions to a non-variational Schr\"odinger system

TL;DR

This paper proves the existence of positive normalized solutions for a class of non-variational coupled Schrödinger systems, using degree theory on a product of $L^2$-spheres.

Contribution

It introduces a novel degree-theoretical approach to establish solutions for non-variational elliptic systems with both cooperative and competitive couplings.

Findings

Existence of positive normalized solutions is established.

The method applies to systems with various coupling types.

Solutions are obtained via operator equations on $L^2$-sphere products.

Abstract

We establish the existence of positive normalized (in the $L^2$ sense) solutions to non-variational weakly coupled elliptic systems of $\ell$ equations. We consider couplings of both cooperative and competitive type. We show the problem can be formulated as an operator equation on the product of $\ell$ $L^2$-spheres and apply a degree-theoretical argument on this product to obtain existence.