Homology of saddle point reduction and applications to resonant elliptic systems

TL;DR

This paper establishes an isomorphism between the critical groups of original and reduced functionals in saddle point reduction, enabling the derivation of multiple solutions for resonant elliptic systems despite lack of Palais-Smale condition.

Contribution

It introduces a novel approach linking critical groups in saddle point reduction, facilitating solutions for resonant elliptic systems with non-compact variational functionals.

Findings

Proves isomorphism of critical groups in saddle point reduction.

Obtains two nontrivial solutions for resonant elliptic systems.

Overcomes lack of Palais-Smale condition using saddle point reduction.

Abstract

In the setting of saddle point reduction, we prove that the critical groups of the original functional and the reduced functional are isomorphic. As application, we obtain two nontrivial solutions for elliptic gradient systems which may be resonant both at the origin and at infinity. The difficulty that the variational functional does not satisfy the Palais-Smale condition is overcame by taking advantage of saddle point reduction. Our abstract results on critical groups are crucial.