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An infinite-dimensional mountain pass theorem with applications to nonlinear elliptic systems

Ablanvi Songo, Fabrice Colin

TL;DR

This work develops an infinite-dimensional generalization of Rabinowitz's Mountain Pass Theorem by leveraging Kryszewski-Szulkin degree theory and a tau-based minimax framework to handle strongly indefinite functionals. It establishes an abstract critical-point theorem and applies it to a semilinear elliptic system in $\mathbb{R}^2$ with subcritical exponential growth, overcoming lack of compactness via Trudinger–Moser control. The main contributions include the new generalized mountain pass result and its application guaranteeing at least one nontrivial solution to a strongly indefinite Hamiltonian system. The approach integrates degree theory, variational methods, and exponential-growth estimates to extend finite-dimensional mountain pass theory to infinite-dimensional, indefinite settings with unbounded domains.

Abstract

The purpose of this paper is to establish a critical point theorem, which is an infinite-dimensional generalization of the classical generalized Mountain Pass Theorem of P. H. Rabinowitz \cite[Theorem 5.3]{Ra}. As application, we obtain the existence of at least one solution to a semilinear elliptic systems with indefinite weights in $\mathbb{R}^2$.

An infinite-dimensional mountain pass theorem with applications to nonlinear elliptic systems

TL;DR

This work develops an infinite-dimensional generalization of Rabinowitz's Mountain Pass Theorem by leveraging Kryszewski-Szulkin degree theory and a tau-based minimax framework to handle strongly indefinite functionals. It establishes an abstract critical-point theorem and applies it to a semilinear elliptic system in with subcritical exponential growth, overcoming lack of compactness via Trudinger–Moser control. The main contributions include the new generalized mountain pass result and its application guaranteeing at least one nontrivial solution to a strongly indefinite Hamiltonian system. The approach integrates degree theory, variational methods, and exponential-growth estimates to extend finite-dimensional mountain pass theory to infinite-dimensional, indefinite settings with unbounded domains.

Abstract

The purpose of this paper is to establish a critical point theorem, which is an infinite-dimensional generalization of the classical generalized Mountain Pass Theorem of P. H. Rabinowitz \cite[Theorem 5.3]{Ra}. As application, we obtain the existence of at least one solution to a semilinear elliptic systems with indefinite weights in .

Paper Structure

This paper contains 10 sections, 10 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Kryszewski-Szulkin degree theory and a general minimax principle
  3. Kryszewski-Szulkin degree theory
  4. A general minimax principle
  5. A new generalized Mountain Pass Theorem
  6. The main result
  7. Proof of Theorem $\ref{['theorem 3.1']}$
  8. Application
  9. An existence result
  10. Proof of Theorem $\ref{['theo 4.1']}$

Key Result

Theorem 1.1

Let $(E, \|\cdot\|)$ be a real Banach space with $E=V\oplus W$, where $V$ is finite dimensional. Suppose $I\in\mathcal{C}^1(E, \mathbb{R})$, satisfies $(PS)$, that is, any sequence $(u_m) \subset E$ for which $I(u_m)$ is bounded and $I'(u_m)\to 0$ as $m\to \infty$ possesses a convergent subsequence. Then $I$ possesses a critical value $c\ge \alpha$ which can be characterized as where

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Theorem 2.1: General minimax principle
  • Corollary 2.1
  • Remark 3
  • Theorem 3.1: Generalized Mountain Pass Theorem
  • Remark 4
  • proof
  • Remark 5
  • ...and 11 more