In this paper, we study the existence of solution for some p(x)-polyharmonic Kirchhoff equations. The latter is allowed to vanish at the origin (degenerate case). Firstly, we study the existence of solutions of approximate equations. Secondly, we prove the existence of the solutions of the original equation. The main tool is the Schauder’s Theorem.
References
[1]
Kratochvil, A. and Necas, J. (1971) The Discreteness of the Spectrum of a Nonlinear Strum-Liouville Equation of Fourth Order. Commentationes Mathematicae Universitatis Carolinae, 12, 639-653.
[2]
Drabek, P. and Otani, M. (2001) Global Bifurcation Result for the p-Biharmonic Operator. Electronic Journal of Differential Equations, 2001, 1-19.
[3]
El Khalil, A., Kellati, S. and Touzani, A. (2002) On the Spectrum of the p-Biharmonic Operator. Electronic Journal of Differential Equations, 9, 161-170.
[4]
Correa, F.I.S.A. and Fijueiredo, G.M. (2006) On an Elliptic Equation of p-Kirchhoff Type via Variational Methods. Bulletin of the Australian Mathematical Society 74, 236-277. https://doi.org/10.1017/S000497270003570X
[5]
Ma, T.F. (2005) Remarks on an Elliptic Equation of Kirchhoff Type. Nonlinear Analysis, 63, 1967-1977. https://doi.org/10.1016/j.na.2005.03.021
[6]
Ma, T.F. (2005) Positive Solutions for a Nonlinear Kirchhoff Type Beam Equation. Applied Mathematics Letters, 18, 479-482.
https://doi.org/10.1016/j.aml.2004.03.013
[7]
Autuori, G., Pucci, P. and Salvatori, M.C. (2009) Asymptotic Stability for Nonlinear Kirchhoff Systems. Nonlinear Analysis, 10, 889-809.
https://doi.org/10.1016/j.nonrwa.2007.11.011
[8]
Villaggio, P. 1997 Mathmatical Models for Elastic Structures. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511529665
[9]
Cavalcanti, M.M., Domingos Cavalcanti, V.N. and Soriano, J.A. (2001) Global Existence and Uniform Decay Rates for the Kiechhoff-Carrier Equation with Nonlinear Dissipation. Advances in Differential Equations, 6, 701-730.
[10]
D’Ancona, P. and Spagnolo, S. (1992) Global Solvability for the Degenerate Kirchhoff Equation with Real Analytic Data. Inventiones Mathematicae, 108, 247-262.
https://doi.org/10.1007/BF02100605
[11]
Dai, G. and Hao, R. (2009) Existence of Solutions for a p(x)-Kirchhoff-Type Equation. Journal of Mathematical Analysis and Applications, 359, 275-284.
https://doi.org/10.1016/j.jmaa.2009.05.031
[12]
Dai, G. and Wei, J. (2010) Infinitely Many Non-Negative Solutions for a p(x)-Kirchhoff-Type Problem with Dirchlet Boundary Condition. Nonlinear Analysis: Theory, Methods & Applications, 73, 3420-3430.
https://doi.org/10.1016/j.na.2010.07.029
[13]
Benilan, P., Brezis, H. and Crandall, M.G. (1975) A Semilinear Equation in L1(RN). Annali della Scuola Normale Superiore di Pisa, 2, 523-555.
[14]
Boccardo, L., Murat, F. and Puel, J.P. (1992) L∞-Estimate for Nonlinear Elliptic Partial Differential Equations and Application to an Existence Result. SIAM Journal on Mathematical Analysis, 23, 326-333. https://doi.org/10.1137/0523016
[15]
Arcoya, D. and Boccardo, L. (2015) Regularizing Effect of the Interplay between Coefficients in Some Epllitic Equations. Journal of Functional Analysis, 268, 1153-1166.
https://doi.org/10.1016/j.jfa.2014.11.011
[16]
Arcoya, D. and Boccardo, L. (2017) Regularizing Effect of Lq Interplay between Coefficients in Some Epllitic Equations. Journal de Mathématiques Pures et Appliquées, 111, 106-125. https://doi.org/10.1016/j.matpur.2017.08.001
[17]
Colasuonno, F. and Pucci, P. (2011) Multiplicity of Solutions for p(x)-Polyharmonic Elliptic Kirchhoff Equations. Nonlinear Analysis, 74, 5962-5974.
https://doi.org/10.1016/j.na.2011.05.073
[18]
Adams, R.A. and Fournier, J.J.F. (2003) Sobolev Spaces. In: Pure and Applied Mathematics, 2nd Edition, Springer, Amsterdam.
[19]
Gilbarg, D. and Trudinger, N. (2001) Elliptic Partial Differential Equations of Second Order. In: Classics in Mathematics, Springer, Berlin.
[20]
Gazzola, F., Grunau, H.C. and Sweers, G. (2010) Polyharmonic Boundary Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. In: Lecture Notes in Mathematics, Springer, Berlin.
https://doi.org/10.1007/978-3-642-12245-3
[21]
Boccardo, L. and Croce, G. (2013) Elliptic Partial Differential Equations (Existence and Regularity of Distributional Solutions). De Gruyter, Berlin.
https://doi.org/10.1515/9783110315424
[22]
Dinening, L., Harjulehto, P., Hasto, P. and Ruzicka, M. (2011) Lebesgue and Sobolev Speaces with Variable Exponents. In: Lecture Notes, Springer, Berlin.
https://doi.org/10.1007/978-3-642-18363-8
[23]
Kovacik, O. and Rakosnik, J. (1991) On Spaces Lp(x) and W1,p(x). Czechoslovak Mathematical Journal, 41, 592-618.
[24]
Fan, X.L. and Zhao, D. (2001) On the Spaces Lp(x) and Wm,p(x). Journal of Mathematical Analysis and Applications, 263, 424-446.
https://doi.org/10.1006/jmaa.2000.7617
[25]
Harjulehto, P., Hasto, P., Koskenoja, M. and Varonen, S. (2006) The Dirchlet Energy Integral and Variable Exponents Sobolev Speaces with Zero Boundary Values, Potential Analysis, 25, 205-222. https://doi.org/10.1007/s11118-006-9023-3
[26]
Diening, L. and Ruzicka, M. (2003) Calderon-Zygmund Operators on Generalized Lebesgue Spaces Lp(·) and Problems Related to Fluid Dynamics. Journal für die reine und angewandte Mathematik, 563, 197-220.
https://doi.org/10.1515/crll.2003.081
[27]
Dinening, L. (2004) Riesz Potential and Sobolev Embeddings on Generalized Lebesdue and Sobolev Speaces Lp(·) and Wk,p(·). Mathematische Nachrichten, 268, 31-43. https://doi.org/10.1002/mana.200310157
