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Normalized solutions for a class of Sobolev critical Schrodinger systems

Houwang Li, Tianhao Liu, Wenming Zou

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.

Normalized solutions for a class of Sobolev critical Schrodinger 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.

Paper Structure

This paper contains 12 sections, 23 theorems, 182 equations.

Table of Contents

  1. Introduction and statement of results
  2. -subcritical
  3. -critical and -supercritical
  4. Sobolev critical
  5. Notations
  6. Structure of the paper
  7. Existence for the attractive case
  8. The case
  9. The case
  10. Existence for the repulsive case
  11. Nonexistence for the defocusing case and
  12. The Sobolev critical case

Key Result

Theorem 1.1

Let $N\geq 3$, $\omega_1<0$ and $\omega_2<0$. Assuming that the exponents satisfy $2<p< 2^*$ and $\alpha+\beta=2^*$ with then we have the following

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.5
  • Remark 1.4
  • ...and 42 more