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Concentration-compactness via profile decomposition for systems of coupled Schrödinger equations of Hamiltonian type

Anderson Cardoso, João Marcos do Ó, Diego Ferraz

Abstract

We analyse Hamiltonian-type systems of second-order elliptic PDE invariant under a non-compact group and, consequently, involve a lack of compactness of the Sobolev embedding. We show that the loss of compactness can be compensated by using a concentration-compactness principle via weak profile decomposition for bounded Palais-Smale sequences in Banach spaces. Our analysis to prove the existence of ground states involves a reduction by the inversion method of the system to a fourth-order equation combined with a variational principle of a minimax nature. Among other results, including regularity and a Pohozaev-type identity, we also prove the non-existence of weak solutions for a class of Lane-Emden systems.

Concentration-compactness via profile decomposition for systems of coupled Schrödinger equations of Hamiltonian type

Abstract

We analyse Hamiltonian-type systems of second-order elliptic PDE invariant under a non-compact group and, consequently, involve a lack of compactness of the Sobolev embedding. We show that the loss of compactness can be compensated by using a concentration-compactness principle via weak profile decomposition for bounded Palais-Smale sequences in Banach spaces. Our analysis to prove the existence of ground states involves a reduction by the inversion method of the system to a fourth-order equation combined with a variational principle of a minimax nature. Among other results, including regularity and a Pohozaev-type identity, we also prove the non-existence of weak solutions for a class of Lane-Emden systems.
Paper Structure (24 sections, 31 theorems, 182 equations)

This paper contains 24 sections, 31 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Terminology and preliminaries definitions
  3. Subcritical case: (\ref{['HYPsub']}).
  4. Existence results:
  5. Regularity results:
  6. Critical case: (\ref{['HYPcrit']})
  7. Existence results
  8. Regularity results
  9. Nonexistence results
  10. Additional remarks on the hypotheses
  11. Outline
  12. Preliminaries results
  13. Variational Settings
  14. Existence of bounded Palais-Smale sequences at minimax levels
  15. Self-similar functions and some properties
  16. ...and 9 more sections

Key Result

Theorem 1.6

Assume V_sirakov, V_pesocerto, bem_def--posi_algum, g_dois, g_tres, and that $b(x),$$V(x)$$f(x,t)$ are $\mathbb{Z}^N$--periodic. Then,

Theorems & Definitions (70)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 60 more