In this paper, we focus on Kirchhoff type problems driven by a logarithmic double phase operator with variable exponents. Under very general assumptions on the nonlinearity and using variational tools, like the mountain pass theorem, we establish the existence of at least one nontrivial weak solution for the problem under consideration. Then, under different hypotheses on the reaction term, we are also able to derive a multiplicity result of solutions for our problem. We stress that in order to produce such a multiplicity result a key role is played by a variant of the symmetric mountain pass theorem.
AlvesC.O.FigueiredoG.M., Nonlinear perturbations of a periodic Kirchhoff equation in , Nonlinear Anal.75(5) (2012), 2750–2759. doi:10.1016/j.na.2011.11.017.
2.
AroraR.Crespo-BlancoA.WinkertP., On logarithmic double phase problems. arXiv:2309.09174v1.
3.
AroraR.FiscellaA.MukherjeeT.WinkertP., On critical double phase Kirchhoff problems with singular nonlinearity, Rend. Circ. Mat. Palermo, II. Ser71 (2022), 1079–1106. doi:10.1007/s12215-022-00762-7.
4.
AutuoriG.PucciP.SalvatoriM., Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal.196(2) (2010), 489–516. doi:10.1007/s00205-009-0241-x.
5.
BaroniP.ColomboM.MingioneG., Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations57(2) (2018), 62, 48 pp.
6.
BuenoH.ErcoleG.FerreiraW.ZumpanoA., Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient, J. Math. Anal. Appl.343(1) (2008), 151–158. doi:10.1016/j.jmaa.2008.01.001.
7.
Cabanillas LapaE., Global solutions for a nonlinear Kirchhoff type equation with viscosity, Opuscula Math.43(5) (2023), 689–701. doi:10.7494/OpMath.2023.43.5.689.
8.
CorrêaF.J.S.A., On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal.59(7) (2004), 1147–1155. doi:10.1016/j.na.2004.08.010.
9.
CorrêaF.J.S.A.FigueiredoG.M., A variational approach for a nonlocal and nonvariational elliptic problem, J. Integral Equations Applications22 (2010), 549–557.
10.
FiscellaA.PinamontiA., Existence and multiplicity results for Kirchhoff type problems on a double phase setting, Mediterr. J. Math.20(1) (2023), 33.
11.
FuchsM.MingioneG., Full -regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth, Manuscripta Math.102(2) (2000), 227–250. doi:10.1007/s002291020227.
12.
GaoY.KimY.-H.ZhenM., Existence and classification of positive solutions for coupled purely critical Kirchhoff system, Bull. Math. Sci.14(2) (2024), 2450002. doi:10.1142/S1664360724500024.
13.
GuptaS.DwivediG., Ground state solution to N-Kirchhoff equation with critical exponential growth and without Ambrosetti–Rabinowitz condition, Rend. Circ. Mat. Palermo, II. Ser73 (2024), 45–56. doi:10.1007/s12215-023-00902-7.
HoK.WinkertP., Infinitely many solutions to Kirchhoff double phase problems with variable exponents, Appl. Math. Lett.145 (2023), 108783, 8 pp. doi:10.1016/j.aml.2023.108783.
16.
LiuW.DaiG., Existence and multiplicity results for double phase problem, J. Differ. Equ.265 (2018), 4311–4334. doi:10.1016/j.jde.2018.06.006.
17.
MarcelliniP.PapiG., Nonlinear elliptic systems with general growth, J. Differential Equations221(2) (2006), 412–443. doi:10.1016/j.jde.2004.11.011.
18.
MotreanuD.MotreanuV.PapageorgiouN.S., Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
19.
MusielakJ., Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin, 1983.
20.
PapageorgiouN.S.WinkertP., Applied Nonlinear Functional Analysis, De Gruyter, Berlin, 2018.
21.
SereginG.A.FrehseJ., Regularity of solutions to variational problems of the deformation theory of plasticity with logarithmic hardening, in: Proceedings of the St. Petersburg Math Ematical Society, Vol. V, Amer. Math. Soc., Providence, RI 193, 1999, pp. 127–152.
22.
SunX.SongY.LiangS., On the critical Choquard–Kirchhoff problem on the Heisenberg group, Adv. Nonlinear Anal.12(1) (2023), 210–236. doi:10.1515/anona-2022-0270.
23.
SunX.SongY.LiangS.ZhangB., Critical Kirchhoff equations involving the p-sub-Laplacians operators on the Heisenberg group, Bull. Math. Sci.13(2) (2023), 2250006, 26 pp. doi:10.1142/S1664360722500060.
24.
VetroC., Variable exponent -Kirchhoff type problem with convection, J. Math. Anal. Appl.506 (2022), 125721. doi:10.1016/j.jmaa.2021.125721.
XiangM.ZhangB.RădulescuV.D., Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian, Nonlinearity29(10) (2016), 3186–3205. doi:10.1088/0951-7715/29/10/3186.
