Nontrivial Solutions of Dirac-Laplace Equation on Compact Spin Manifolds

TL;DR

This paper demonstrates the existence of multiple solutions for a nonlinear Dirac-Laplace equation on compact spin manifolds by applying the Fountain theorem and relaxing traditional growth conditions.

Contribution

It introduces a super-quadratic condition replacing the Ambrosetti-Rabinowitz condition, enabling the use of the Fountain theorem to find multiple solutions.

Findings

Multiple solutions are established for the nonlinear Dirac-Laplace equation.

The super-quadratic condition suffices for the Cerami condition.

The approach relaxes traditional growth conditions in nonlinear analysis.

Abstract

We apply the Fountain theorem to a class of nonlinear Dirac-Laplace equation on compact spin manifold. We show the standard Ambrosetti-Rabinowitz condition can be replaced by a more natural super-quadratic condition that is enough to obtain the Cerami condition under certain conditions. Multiple solutions of nonlinear Dirac-Laplace equation are obtained in this note.