Existence and multiplicity of solutions for m-polyharmonic Kirchhoff problems without Ambrosetti–Rabinowitz conditions

Abstract

AbstractIn this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic m-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, we introduce a positive quantity λM similar to the first eigenvalue of the m-polyharmonic operator to find a mountain pass solution, and also to discuss the sublinear case under large growth conditions at infinity and at zero. Our results are an improvement and generalization of the corresponding results obtained by Colasuonno-Pucci (Nonlinear Analysis: Theory, Methods and Applications, 2011) and Bae-Kim (Mathematical Methods in the Applied Sciences, 2020).Keywords: m-Polyharmonic operatorPalais-Smale conditionsymmetric mountain pass theoremschauder basisKrasnoselskii genus theoryKirchhoff equationsCOMMUNICATED BY: A. MezianiAMS Subject Classifications: Primary: 35J5535J65Secondary: 35B65 Disclosure statementNo potential conflict of interest was reported by the author(s).Correction StatementThis article has been republished with minor changes. These changes do not impact the academic content of the article.Notes1 More precisely, Pj are uniformly bounded, that is there exists C>0 such that ‖Pj(z)‖≤C‖z‖ for each j∈N∗ and all z∈W0r,m(Ω) (see [Citation24, Citation26])Additional informationFundingA. Harrabi gratefully acknowledge the approval and the support of this research study by the grant no. SCIA-2022-11-1398 from the Deanship of Scientific Research at Northern Border University, Arar, K.S.A. M. K. Hamdani was supported by the Tunisian Military Research Center for Science and Technology Laboratory LR19DN01. M.K. Hamdani expresses his deepest gratitude to the Military Aeronautical Specialities School, Sfax (ESA) for providing an excellent atmosphere for work. A. Fiscella is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). A. Fiscella realized the manuscript within the auspices of the INdAM-GNAMPA project titled 'Equazioni differenziali alle derivate parziali in fenomeni non lineari' (CUP_E55F22000270001) and of the FAPESP Thematic Project titled 'Systems and partial differential equations' (2019/02512-5). The authors wish to thank Professors Dong Ye and Nguyen Thanh Chung for stimulating discussions on the subject.