Quasilinear problems without the Ambrosetti-Rabinowitz condition

TL;DR

This paper proves the existence of solutions for certain complex quasilinear equations where the operators depend on the unknown, using variational methods and a relaxed compactness condition, without the usual Ambrosetti-Rabinowitz assumption.

Contribution

It introduces a new approach to establish solutions for quasilinear problems without relying on the Ambrosetti-Rabinowitz condition, broadening the scope of solvable equations.

Findings

Existence of nontrivial solutions demonstrated

Applicable to highly quasilinear problems with operator dependence on unknown

Utilizes a weak Cerami-Palais-Smale condition

Abstract

We show the existence of nontrivial solutions for a class of highly quasilinear problems in which the governing operators depend on the unknown function. By using a suitable variational setting and a weak version of the Cerami-Palais-Smale condition, we establish the desired result without assuming that the nonlinear source satisfies the Ambrosetti-Rabinowitz condition.