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Bounded Palais-Smale sequences with Morse type information for some constrained functionals

Jack Borthwick, Xiaojun Chang, Louis Jeanjean, Nicola Soave

Abstract

In this paper, we study, for functionals having a mountain pass geometry on a constraint, the existence of bounded Palais-Smale sequences carrying Morse index type information.

Bounded Palais-Smale sequences with Morse type information for some constrained functionals

Abstract

In this paper, we study, for functionals having a mountain pass geometry on a constraint, the existence of bounded Palais-Smale sequences carrying Morse index type information.
Paper Structure (9 sections, 28 theorems, 192 equations)

This paper contains 9 sections, 28 theorems, 192 equations.

Table of Contents

  1. Introduction and main results
  2. Statement and proof of Theorem \ref{['M-theorem']}
  3. The geodesic spray coming from the scalar product of $H$
  4. Geodesics and $D^2\phi$
  5. Transporting $W$ to other points
  6. Fang-Ghoussoub's result
  7. Proof of Theorem \ref{['thm: application']}
  8. The case of $\mathbb{Z}_2$-homotopic families
  9. Adapting the deformations and proof of Theorem \ref{['M-theoremSym']}

Key Result

Theorem 1.5

Let $I \subset (0,+\infty)$ be an interval and consider a family of $C^2$ functionals $\Phi_\rho: E \to \mathbb{R}$ of the form where $B(u) \ge 0$ for every $u \in E$, and Suppose moreover that $\Phi_\rho'$ and $\Phi_\rho"$ are $\alpha$-Hölder continuous on bounded sets in the sense of Definition Holder continuous (hol2) for some $\alpha \in (0,1]$. Finally, suppose that there exist $w_1, w_2 \i

Theorems & Definitions (68)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 58 more