[Paper Review] Some critical point theorems and applications

Guangcun Lu|arXiv (Cornell University)|Feb 10, 2011

Nonlinear Partial Differential Equations26 references3 citations

TL;DR

This paper establishes new critical point theorems for continuously directional differentiable functionals by leveraging novel splitting theorems, enabling weaker assumptions than prior results. It generalizes computations of critical groups and Morse inequalities, particularly for degenerate and sign-changing critical points, and applies them to prove multiple solutions for nonlinear higher-order elliptic equations.

ABSTRACT

This paper is a continuation of \cite{Lu1}. In Part I, applying the new splitting theorems developed therein we generalize previous some results on computations of critical groups and some critical point theorems to weaker versions. In Part II (in progress), they are used to study multiple solutions for nonlinear higher order elliptic equations described in the introduction of \cite{Lu1}.

Motivation & Objective

• To extend previous critical point theorems by relaxing smoothness assumptions from $C^2$ to continuously directional differentiable functionals.
• To generalize computations of critical groups at degenerate and sign-changing critical points under weaker conditions than in earlier works.
• To develop a framework that avoids complex computations of subdifferentials and weak slopes, enhancing applicability in variational problems.
• To apply the refined critical point theory to prove the existence of multiple solutions for nonlinear higher-order elliptic equations.
• To establish homotopy equivalences and isomorphisms in singular homology to relate critical groups to topological invariants of sublevel sets.

Proposed method

• Utilizes new splitting theorems for continuously directional differentiable functionals to replace classical $C^2$ assumptions.
• Applies deformation lemmas and weak/strong slope concepts to analyze critical points via the weak slope $|df|(u)$ and strong slope $| abla f(u)|$.
• Employs the Cerami $(C)_c$ condition and Palais-Smale $(PS)_c$ condition to ensure compactness of critical sequences.
• Uses homotopy equivalences and excision isomorphisms (via Lemma 5.10, 5.11, 5.12) to relate critical groups to relative homology of sublevel sets.
• Applies the isomorphism $C^q({rak L}, u) \cong \bar{H}^q(N \cup \mathcal{L}^{c-\delta}, \mathcal{L}^{c-\delta} \cap N)$ to compute critical groups at isolated critical points.
• Establishes $\bar{H}^q(\mathcal{L}^{c+\delta} \cup W_\varepsilon, \mathcal{L}^{c-\delta} \cup W_\varepsilon) \cong \bigoplus_{i=1}^m C^q({\frak L}, u_i)$ for isolated critical points, linking topology to critical group structure.

Experimental results

Research questions

• RQ1 How can critical point theorems be generalized for functionals that are only continuously directional differentiable, without requiring $C^2$ or $C^1$ smoothness?
• RQ2 What is the structure of critical groups at degenerate and sign-changing critical points under weaker compactness and differentiability assumptions?
• RQ3 Can the computation of critical groups be simplified by avoiding subdifferential and weak slope calculations?
• RQ4 Under what conditions does the existence of multiple non-trivial critical points follow from topological invariants of sublevel sets?
• RQ5 How do homotopy equivalences between relative homology pairs relate to the critical group structure in variational problems?

Key findings

• The paper establishes that $C^q({\frak L}, u) \cong \bar{H}^q({\mathcal{L}^{c+\delta} \cup W_\varepsilon}, {\mathcal{L}^{c-\delta} \cup W_\varepsilon})$ for small $\delta > 0$, linking critical groups to relative homology.
• For a single isolated critical point $u$ at level $c_1$, the critical group satisfies $C^q({\frak L}, u) \cong \delta_{q1} \mathbb{Z}_2$, indicating a nontrivial $1$-dimensional critical group.
• If $c_1 < {\frak L}(\theta)$, then $K^*_c$ contains at least one non-trivial critical point, and $K^*_{c_1} \neq \emptyset$ under the given assumptions.
• The inclusion $({\mathcal{L}^{b} \cup W_\varepsilon}, {\mathcal{L}^{a} \cup W_\varepsilon}) \to ({\mathcal{L}^{d} \cup W_\varepsilon}, {\mathcal{L}^{a} \cup W_\varepsilon})$ is a homotopy equivalence when $K^*_c = \emptyset$ for $c \in [b,d] \setminus \{c\}$, implying topological invariance.
• The critical group computation at $u$ is isomorphic to $\bar{H}^q(N \cup \mathcal{L}^{c-\delta}, \mathcal{L}^{c-\delta} \cap N)$, enabling topological analysis via neighborhood deformation.
• The result $\bar{H}^q(H, W_\varepsilon; \mathbb{Z}_2) \cong \delta_{q1} \mathbb{Z}_2$ confirms the topological complexity of the space, supporting the existence of nontrivial critical points.