Normalized solutions for a class of Sobolev critical Schrodinger systems

TL;DR

This paper investigates the existence, multiplicity, and non-existence of normalized solutions for Sobolev critical coupled Schrödinger systems, providing new results and methods that advance understanding of these complex PDEs.

Contribution

It introduces new existence and multiplicity results for Sobolev critical Schrödinger systems and offers a comprehensive analysis including non-existence in the defocusing case.

Findings

Established conditions for existence and multiplicity of solutions.

Proved non-existence results in the defocusing case.

Provided methods potentially applicable to open problems in Sobolev critical growth systems.

Abstract

This paper focuses on the existence and multiplicity of normalized solutions for the coupled Schrodinger system with Sobolev critical coupling term. We present several existence and multiplicity results under some explicit conditions. Furthermore, we present a non-existence result for the defocusing case. This paper, together with the paper [T. Bartsch, H. W. Li and W. M. Zou. Calc. Var. Partial Differential Equations 62 (2023) ], provides a more comprehensive understanding of normalized solutions for Sobolev critical systems. We believe our methods can also address the open problem of the multiplicity of normalized solutions for Schrodinger systems with Sobolev critical growth, with potential for future development and broader applicability.