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Minimax Theory and its Applications 06 (2021), No. 2, 239--250
Copyright Heldermann Verlag 2021



Quasilinear Problems without the Ambrosetti-Rabinowitz Condition

Anna Maria Candela
Dip. di Matematica, UniversitÓ degli Studi di Bari Aldo Moro, 70125 Bari, Italy
annamaria.candela@uniba.it

Genni Fragnelli
Dip. di Matematica, UniversitÓ degli Studi di Bari Aldo Moro, 70125 Bari, Italy
genni.fragnelli@uniba.it

Dimitri Mugnai
Dip. di Scienze Ecologiche e Biologiche, UniversitÓ degli Studi della Tuscia, 01100 Viterbo, Italy
dimitri.mugnai@unitus.it



We show the existence of nontrivial solutions for a class of 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.

Keywords: Quasilinear equation, weak Cerami-Palais-Smale condition, failure of the Ambrosetti-Rabinowitz condition, p-superlinear problem, subcritical growth.

MSC: 35J92, 35J20, 35J60.

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