Linking Theorems of Local Semiflows on Complete Metric Spaces

TL;DR

This paper develops linking theorems and mountain pass results for local semiflows on complete metric spaces, providing new tools to find invariant sets and solutions for elliptic and parabolic equations without classical conditions.

Contribution

It introduces an alternative approach to detect invariant sets and solutions in dynamical systems using linking theorems and mountain pass results without relying on Conley index or P.S. conditions.

Findings

Existence of recurrent solutions for nonautonomous parabolic equations.

Existence of positive solutions for elliptic equations.

New linking and mountain pass theorems for local semiflows.

Abstract

In this paper we prove some linking theorems and mountain pass type results for dynamical systems in terms of local semiflows on complete metric spaces. Our results provide an alternative approach to detect the existence of compact invariant sets without using the Conley index theory. They can also be applied to variational problems of elliptic equations without verifying the classical P.S. Condition. As an example, we study the resonant problem of the nonautonomous parabolic equation $ u_t-\Delta u-\mu u=f(u)+g(x,t) $ on a bounded domain. The existence of a recurrent solution is proved under some Landesman-Laser type conditions by using an appropriate linking theorem of semiflows. Another example is the elliptic equation $-\Delta u+a(x)u=f(x,u)$ on $R^n$. We prove the existence of positive solutions by applying a mountain pass lemma of semiflows to the parabolic flow of the problem.