International Journal of Nonlinear Analysis and Applications

Existence of three solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential

Document Type : Research Paper

Authors

1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

2 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

Abstract

In this work, we establish existence results for the following fourth-order Kirchhoff-type elliptic problem with Hardy potential
\begin{equation*}
\begin{gathered}
M \Big(\int_{\Omega} |\Delta u|^p dx\Big) \Delta_p^2 u -
\frac{a}{|x|^{p}} |u|^{p-2} u = \lambda f(x, u), \quad \text{in } \Omega, \\
u = \Delta u = 0, \quad \text{on } \partial \Omega.
\end{gathered}
\end{equation*}
Precisely, by using the classical Hardy inequality and critical point theory, we prove the existence of multiple weak solutions for the fourth-order Kirchhoff-type elliptic problem with Hardy potential.

Keywords

[1] C.O. Alves, F.S.J.A. Correa, and T. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93.
[2] E. Berchio, D. Cassani, and F. Gazzola, Hardy-Rellich inequalities with boundary remainder terms and applications, Manuscripta Math. 131 (2010), 427–458.
[3] G. Bonanno, A minimax inequality and its applications to ordinary differential equations, J. Math. Anal. Appl. 270 (2002), 210–229.
[4] G. Bonanno, Multiple critical points theorems without the Palais-Smale condition, J. Math. Anal Appl. 299 (2004), 600–614.
[5] A. Callegari and A. Nachman, A nonlinear singular boundary value problem in the theory of pseudoplastic-fluids, SIAM J. Appl. Math. 38 (1980), 275–281.
[6] P. Candito, L. Li, and R. Livrea, Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the p-biharmonic, Nonlinear Anal. 75 (2012), 6360–6369.
[7] P. Candito and R. Livrea, Infinitely many solutions for a nonlinear Navier boundary value problem involving the p-biharmonic, Studia Univ. Babes-Bolyai Math. 55 (2010), no. 4, 41–51.
[8] P. Candito and G. Molica Bisci, Multiple solutions for a Navier boundary value problem involving the p-biharmonic operator, Discrete Contin. Dyn. Syst. Series S. 5 (2012), 741–751.
[9] B. Cheng, X. Wu, and J. Liu, Multiplicity of solutions for nonlocal elliptic system of (p, q)-Kirchhoff type, Abstr. Appl. Anal. 2011 (2021), Article ID 526026, 13 pages.
[10] N.T. Chung, Multiple solutions for a fourth order elliptic equation with Hardy type potential, Acta Univ. Apulensis Math. 28 (2011, 115–124.
[11] F.S.J.A. Correa and R.G. Nascimento, On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition, Math. Comput. Modelling 49 (2009), 598–604.
[12] M. Ferrara, S. Khademloo, and S. Heidarkhani, Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems, Appl. Math. Comput. 234 (2014), 316–325.
[13] M. Ferrara and G. Molica Bisci, Existence results for elliptic problems with Hardy potential, Bull. Sci. Math. 138 (2014), 846–859.
[14] G.M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013), 706–713. [15] G.P. Garcia Azorero, and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differ. Equ. 144 (1998), 441–476.
[16] M. Ghergu and V. Radulescu, Singular Elliptic Problems, Bifurcation & Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, vol. 37, Oxford University Press, New York, 2008.
[17] M. Ghergu and V. Radulescu, Sublinear singular elliptic problems with two parameters, J. Differ. Equ. 195 (2003), 520–536.
[18] J. R. Graef, S. Heidarkhani, and L. Kong, A variational approach to a Kirchhoff-type problem involving two parameters, Results. Math. 63 (2013), 877–889.
[19] J.R. Graef, S. Heidarkhani, and L. Kong, Multiple solutions for a class of (p1, . . . , pn)-biharmonic systems, Commun. Pure Appl. Anal. 12 (2013), 1393–1406.
[20] A. Hamydy, M. Massar, N. Tsouli, Existence of solutions for p-Kirchhoff type problems with critical exponent, Electron. J. Diff. Equ., 2011 (2011), no. 105, 1–8.
[21] X. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal.70 (2009), 1407–1414.
[22] S. Heidarkhani, Existence of non-trivial solutions for systems of n fourth-order partial differential equations, Math. Slovaca 64 (2014), 1249–1266.
[23] S. Heidarkhani, Infinitely many solutions for systems of n two-point boundary value Kirchhoff-type problems, Ann. Polon. Math. 107.2 (2013) 133-152.
[24] S. Heidarkhani, Non-trivial solutions for a class of (p1, ..., pn)-biharmonic systems with Navier boundary conditions, Ann. Polon. Math. 105.1 (2012), 65–76.
[25] S. Heidarkhani, S. Khademloo, and A. Solimaninia, Multiple solutions for a perturbed fourth-order Kirchhoff type elliptic problem, Portugal. Math. (N.S.) 71 (2014), 39–61.
[26] S. Heidarkhani, Y. Tian, and C.-L. Tang, Existence of three solutions for a class of (p1, ..., pn)-biharmonic systems with Navier boundary conditions, Ann. Polon. Math. 104 (2012), 261–277.
[27] Y. Huang and X. Liu, Sign-changing solutions for p-biharmonic equations with Hardy potential, J. Math. Anal. Appl. 412 (2014), 142-154.
[28] D. Juan and P. Ireneo Nonlinear elliptic problems with a singular weight on the boundary Cal. Variat. 41 (2011), 567–586.
[29] M. Khodabakhshi, A.M. Aminpour, G.A. Afrouzi, and A. Hadjian, Existence of two weak solutions for some singular elliptic problems, Rev. Serie A Mat. (RACSAM) 110 (2016), 385–393.
[30] G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883.
[31] A. Kristaly and C. Varga, Multiple solutions for elliptic problems with singular and sublinear potentials, Proc. Am. Math. Soc. 135 (2007), 2121–2126.
[32] L. Li and S. Heidarkhani, Existence of three solutions to a double eigenvalue problem for the p-biharmonic equation, Ann. Polon. Math. 104 (2012), 71–80.
[33] Y. Li, F. Li, and J. Shi, Existence of a positive solution to Kirchhoff-type problems without compactness conditions, J. Differ. Equ. 253 (2012), 2285–2294.
[34] L. Li and C.-L. Tang, Existence of three solutions for (p, q)-biharmonic systems, Nonlinear Anal. 73 (2010), 796–805.
[35] C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal. 72 (2010), 1339–1347.
[36] J. Liu, S.X. Chen, and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in RN, J. Math. Anal. Appl. 395 (2012), 608–615.
[37] M. Massar, Existence and multiplicity solutions for nonlocal elliptic problems, Electron. J. Diff. Equ. 2013 (2013), no. 75, 1–14.
[38] M. Massar, E.M. Hssini, N. Tsouli, and M. Talbi, Infinitely many solutions for a fourth-order Kirchhoff type elliptic problem, J. Math. Comput. Sci. 8 (2014), 33–51.
[39] G. Molica Bisci and P. Pizzimenti, Sequences of weak solutions for non-local elliptic problems with Dirichlet boundary condition, Proc. Edinb. Math. Soc. 57 (2014), 779–809.
[40] G. Molica Bisci and D. Repovs, Multiple solutions of p-biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equ. 59 (2014), 271–284.
[41] M. Negravi, G.A. Afrouzi, Multiplicity results for perturbed Kirchhoff-type elliptic problems with Hardy potential, preprint.
[42] R. Pei, J. Zhang, Sign-changing solutions for a fourth-order elliptic equation with Hardy singular terms, J. Appl. Math. 2013 (2013), Article ID 627570, 6 pages.
[43] M. Perez-Llanos and A. Primo, Semilinear biharmonic problems with a singular term, J. Diff Equ. 257 (2014), 3200–3225.
[44] V. Radulescu, Combined effects in nonlinear singular elliptic problems with convention, Rev. Roum. Math. Pures Appl. 53 (2008), 543–553.
[45] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (2010), 543–549.
[46] F. Wang and Y. An, Existence and multiplicity of solutions for a fourth-order elliptic equation, Bound. Value Probl. 2012 (2012), 1–9.
[47] Y. Wang and Y. Shen, Nonlinear biharmonic equations with Hardy potential and critical parameter, J. Math. Anal. Appl. 355 (2009), 649–660.
[48] H. Xie and J. Wang, Infinitely many solutions for p-harmonic equation with singular term, J. Inequal. Appl. 2013 (2013), no. 9, 1–13.
[49] M. Xu and C. Bai, Existence of infinitely many solutions for perturbed Kirchhoff type elliptic problems with Hardy potential, Electron. J. Diff. Equ. 2015 (2015), no. 268, 1–9.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 5
May 2024
Pages 11-22
  • Receive Date: 19 February 2023
  • Accept Date: 28 April 2023