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January  2016, 15(1): 91-102. doi: 10.3934/cpaa.2016.15.91

## Multiple nontrivial solutions to a $p$-Kirchhoff equation

 1 School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China 2 School of Mathematical Sciences, Capital Normal University, Beijing 100048

Received  January 2015 Revised  April 2015 Published  December 2015

In this paper, by computing the relevant critical groups, we obtain nontrivial solutions via Morse theory to the nonlocal $p$-Kirchhoff-type quasilinear elliptic equation \begin{eqnarray} (P)\quad\quad &&\displaystyle\bigg[M\bigg(\int_\Omega|\nabla u|^p dx\bigg)\bigg]^{p-1}(-\Delta_pu) = f(x,u), \quad x\in\Omega,\\ && u=0, \quad x\in \partial \Omega, \end{eqnarray} where $\Omega \subset \mathbb R^N$ is a bounded open domain with smooth boundary $\partial \Omega$ and $N \geq 3$.
Citation: Anran Li, Jiabao Su. Multiple nontrivial solutions to a $p$-Kirchhoff equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 91-102. doi: 10.3934/cpaa.2016.15.91
##### References:
 [1] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar [2] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string,, Trans. Amer. Math. Soc., 348 (1996), 305. doi: 10.1090/S0002-9947-96-01532-2. Google Scholar [3] T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,, Nonlinear Anal., 28 (1997), 419. doi: 10.1016/0362-546X(95)00167-T. Google Scholar [4] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems,, first ed., (1993). doi: 10.1007/978-1-4612-0385-8. Google Scholar [5] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems,, Nonlinear Anal., 30 (1997), 4619. doi: 10.1016/S0362-546X(97)00169-7. Google Scholar [6] F. J. S. A. Corrêa and G. M. Figueiredo, On a elliptic equation of $p$-Kirchhoff type via variational methods,, Bull. Aust. Math. Soc., 74 (2006), 263. doi: 10.1017/S000497270003570X. Google Scholar [7] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data,, Invent. Math., 108 (1992), 247. doi: 10.1007/BF02100605. Google Scholar [8] M. Dreher, The Kirchhoff equation for the $p$-Laplacian,, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), 217. Google Scholar [9] M. Dreher, The ware equation for the $p$-Laplacian,, Hokkaido Math. J., 36 (2007), 21. doi: 10.14492/hokmj/1285766660. Google Scholar [10] Q. Jiu and J. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian,, J. Math. Anal. Appl., 281 (2003), 587. doi: 10.1016/S0022-247X(03)00165-3. Google Scholar [11] G. Kirchhoff, Mechanik,, Teubner, (1883). Google Scholar [12] S. Li and J. Liu, Some existence theorems on multiple critical points and their applications,, Kexue Tongbao, 17 (1984), 1025. Google Scholar [13] P. Lindqvist, On the equation div$(|\nabla u|^{p-2}\nabla u) + \lambda|u|^{p-2}u=0$,, Proc. Amer. Math. Soc., 109 (1990), 157. doi: 10.2307/2048375. Google Scholar [14] J. L. Lions, On some equations in boundary value problems of mathematical physics,, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, (1997), 284. Google Scholar [15] D. Liu and P. Zhao, Multiple nontrivial solutions to a $p$-Kirchhoff equation,, Nonlinear Anal, 75 (2012), 5032. doi: 10.1016/j.na.2012.04.018. Google Scholar [16] J. Liu, A Morse index for a saddle point,, Syst. Sc. Math. Sc., 2 (1989), 32. Google Scholar [17] S. Liu, Existence of solutions to a superlinear p-Laplacian equation,, Electron. J. Differential Equations, 66 (2001), 1. Google Scholar [18] J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems,, J. Math. Anal. Appl., 258 (2001), 209. doi: 10.1006/jmaa.2000.7374. Google Scholar [19] A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,, Nonlinear Anal., 70 (2009), 1275. doi: 10.1016/j.na.2008.02.011. Google Scholar [20] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, first ed., (1989). doi: 10.1007/978-1-4757-2061-7. Google Scholar [21] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006. Google Scholar [22] J. Su, Multiplicity results for asymptotically linear elliptic problems at resonance,, J. Math. Anal. Appl., 278 (2003), 397. doi: 10.1016/S0022-247X(02)00707-2. Google Scholar [23] J. Sun and S. Liu, Nontrivial solutions of Kirchhoff type problems,, Applied Mathematics Letters, 25 (2012), 500. doi: 10.1016/j.aml.2011.09.045. Google Scholar [24] Z. Q. Wang, On a superlinear elliptic equation,, Ann. Inst. H. Poincaré, 8 (1991), 43. Google Scholar [25] Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, J. Math. Anal. Appl., 317 (2006), 456. doi: 10.1016/j.jmaa.2005.06.102. Google Scholar

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##### References:
 [1] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar [2] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string,, Trans. Amer. Math. Soc., 348 (1996), 305. doi: 10.1090/S0002-9947-96-01532-2. Google Scholar [3] T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,, Nonlinear Anal., 28 (1997), 419. doi: 10.1016/0362-546X(95)00167-T. Google Scholar [4] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems,, first ed., (1993). doi: 10.1007/978-1-4612-0385-8. Google Scholar [5] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems,, Nonlinear Anal., 30 (1997), 4619. doi: 10.1016/S0362-546X(97)00169-7. Google Scholar [6] F. J. S. A. Corrêa and G. M. Figueiredo, On a elliptic equation of $p$-Kirchhoff type via variational methods,, Bull. Aust. Math. Soc., 74 (2006), 263. doi: 10.1017/S000497270003570X. Google Scholar [7] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data,, Invent. Math., 108 (1992), 247. doi: 10.1007/BF02100605. Google Scholar [8] M. Dreher, The Kirchhoff equation for the $p$-Laplacian,, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), 217. Google Scholar [9] M. Dreher, The ware equation for the $p$-Laplacian,, Hokkaido Math. J., 36 (2007), 21. doi: 10.14492/hokmj/1285766660. Google Scholar [10] Q. Jiu and J. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian,, J. Math. Anal. Appl., 281 (2003), 587. doi: 10.1016/S0022-247X(03)00165-3. Google Scholar [11] G. Kirchhoff, Mechanik,, Teubner, (1883). Google Scholar [12] S. Li and J. Liu, Some existence theorems on multiple critical points and their applications,, Kexue Tongbao, 17 (1984), 1025. Google Scholar [13] P. Lindqvist, On the equation div$(|\nabla u|^{p-2}\nabla u) + \lambda|u|^{p-2}u=0$,, Proc. Amer. Math. Soc., 109 (1990), 157. doi: 10.2307/2048375. Google Scholar [14] J. L. Lions, On some equations in boundary value problems of mathematical physics,, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, (1997), 284. Google Scholar [15] D. Liu and P. Zhao, Multiple nontrivial solutions to a $p$-Kirchhoff equation,, Nonlinear Anal, 75 (2012), 5032. doi: 10.1016/j.na.2012.04.018. Google Scholar [16] J. Liu, A Morse index for a saddle point,, Syst. Sc. Math. Sc., 2 (1989), 32. Google Scholar [17] S. Liu, Existence of solutions to a superlinear p-Laplacian equation,, Electron. J. Differential Equations, 66 (2001), 1. Google Scholar [18] J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems,, J. Math. Anal. Appl., 258 (2001), 209. doi: 10.1006/jmaa.2000.7374. Google Scholar [19] A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,, Nonlinear Anal., 70 (2009), 1275. doi: 10.1016/j.na.2008.02.011. Google Scholar [20] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, first ed., (1989). doi: 10.1007/978-1-4757-2061-7. Google Scholar [21] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006. Google Scholar [22] J. Su, Multiplicity results for asymptotically linear elliptic problems at resonance,, J. Math. Anal. Appl., 278 (2003), 397. doi: 10.1016/S0022-247X(02)00707-2. Google Scholar [23] J. Sun and S. Liu, Nontrivial solutions of Kirchhoff type problems,, Applied Mathematics Letters, 25 (2012), 500. doi: 10.1016/j.aml.2011.09.045. Google Scholar [24] Z. Q. Wang, On a superlinear elliptic equation,, Ann. Inst. H. Poincaré, 8 (1991), 43. Google Scholar [25] Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, J. Math. Anal. Appl., 317 (2006), 456. doi: 10.1016/j.jmaa.2005.06.102. Google Scholar
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