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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.

[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.

[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.

[4]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems,, first ed., (1993). doi: 10.1007/978-1-4612-0385-8.

[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.

[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.

[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.

[8]

M. Dreher, The Kirchhoff equation for the $p$-Laplacian,, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), 217.

[9]

M. Dreher, The ware equation for the $p$-Laplacian,, Hokkaido Math. J., 36 (2007), 21. doi: 10.14492/hokmj/1285766660.

[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.

[11]

G. Kirchhoff, Mechanik,, Teubner, (1883).

[12]

S. Li and J. Liu, Some existence theorems on multiple critical points and their applications,, Kexue Tongbao, 17 (1984), 1025.

[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.

[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.

[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.

[16]

J. Liu, A Morse index for a saddle point,, Syst. Sc. Math. Sc., 2 (1989), 32.

[17]

S. Liu, Existence of solutions to a superlinear p-Laplacian equation,, Electron. J. Differential Equations, 66 (2001), 1.

[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.

[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.

[20]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, first ed., (1989). doi: 10.1007/978-1-4757-2061-7.

[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.

[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.

[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.

[24]

Z. Q. Wang, On a superlinear elliptic equation,, Ann. Inst. H. Poincaré, 8 (1991), 43.

[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.

show all references

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.

[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.

[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.

[4]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems,, first ed., (1993). doi: 10.1007/978-1-4612-0385-8.

[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.

[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.

[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.

[8]

M. Dreher, The Kirchhoff equation for the $p$-Laplacian,, Rend. Semin. Mat. Univ. Politec. Torino, 64 (2006), 217.

[9]

M. Dreher, The ware equation for the $p$-Laplacian,, Hokkaido Math. J., 36 (2007), 21. doi: 10.14492/hokmj/1285766660.

[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.

[11]

G. Kirchhoff, Mechanik,, Teubner, (1883).

[12]

S. Li and J. Liu, Some existence theorems on multiple critical points and their applications,, Kexue Tongbao, 17 (1984), 1025.

[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.

[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.

[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.

[16]

J. Liu, A Morse index for a saddle point,, Syst. Sc. Math. Sc., 2 (1989), 32.

[17]

S. Liu, Existence of solutions to a superlinear p-Laplacian equation,, Electron. J. Differential Equations, 66 (2001), 1.

[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.

[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.

[20]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, first ed., (1989). doi: 10.1007/978-1-4757-2061-7.

[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.

[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.

[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.

[24]

Z. Q. Wang, On a superlinear elliptic equation,, Ann. Inst. H. Poincaré, 8 (1991), 43.

[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.

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