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

TL;DR

This paper extends the classical Mountain Pass Theorem to infinite-dimensional spaces and applies it to prove the existence of solutions for certain nonlinear elliptic systems with indefinite weights.

Contribution

It introduces a new infinite-dimensional critical point theorem and demonstrates its application to nonlinear elliptic systems with indefinite weights.

Findings

Established an infinite-dimensional critical point theorem.

Proved existence of solutions for semilinear elliptic systems with indefinite weights.

Generalized classical mountain pass results to broader settings.

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 $\mathbb{R}^2$.