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November  2016, 15(6): 2161-2177. doi: 10.3934/cpaa.2016032

Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type

Wen Zhang 1, , Xianhua Tang 2, , Bitao Cheng 3, and  Jian Zhang 1,

1. 

School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, China
2. 

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083
3. 

School of Mathematics and Statistics, Qujing Normal University, Qujing, 655011 Yunnan, China

Received  December 2015 Revised  March 2016 Published  September 2016

In this paper, we study the following fourth-order elliptic equation with Kirchhoff-type \begin{eqnarray} \left\{\begin{array}{l} \Delta^{2}u-(a+b\int_{\mathbb{R}^N}|\nabla u|^{2}dx)\Delta u+V(x)u=f(u),\ \ \ x\in \mathbb{R}^{N},\\ u\in H^{2}(\mathbb{R}^{N}), \end{array}\right. \end{eqnarray} where the constants $a>0, b\geq 0$. By constraint variational method and quantitative deformation lemma, we obtain that the problem possesses one least energy sign-changing solution $u_{b}$. Moreover, we also prove that the energy of $u_{b}$ is strictly larger than two times the ground state energy. Finally, we give a convergence property of $u_{b}$ when $b$ as a parameter and $b\rightarrow 0$.
Keywords: nonlocal term., sign-changing solutions, Fourth order elliptic equations, Kirchhoff-type.
Mathematics Subject Classification: 35J50, 35J6.
Citation: Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2161-2177. doi: 10.3934/cpaa.2016032
References:
[1]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 1153.  doi: 10.1007/s00033-013-0376-3.  Google Scholar

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J. M. Ball, Initial-boundary value for an extensible beam,, \emph{J. Math. Anal. Appl.}, 42 (1973), 61.   Google Scholar

[3]

H. M. Berger, A new approach to the analysis of large deflections of plates,, \emph{J. Appl. Mech.}, 22 (1955), 465.   Google Scholar

[4]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$,, \emph{Commun. Partial Differ. Equ.}, 20 (1995), 1725.  doi: 10.1080/03605309508821149.  Google Scholar

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T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology,, \emph{Ann. I. H. Poincar\'e Anal, 22 (2005), 259.  doi: 10.1016/j.anihpc.2004.07.005.  Google Scholar

[6]

T. Bartch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems,, \emph{J. Anal. Math.}, 96 (2005), 1.  doi: 10.1007/BF02787822.  Google Scholar

[7]

T. Bartsch, Z. Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations,, in \emph{Handbook of Differential Equations-Stationary Partial Differential Equations} (M. Chipot and P. Quittner eds.), (2005), 1.  doi: 10.1016/S1874-5733(05)80009-9.  Google Scholar

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B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, \emph{Nonlinear Anal.}, 71 (2009), 4883.  doi: 10.1016/j.na.2009.03.065.  Google Scholar

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B. T. Cheng, X. Wu and J. Liu, Multiple solutions for a class of Kirchhoff-type problems with concave nonlinearity,, \emph{NoDEA Nonl. Diff. Equa. Appl.}, 19 (2012), 521.  doi: 10.1007/s00030-011-0141-2.  Google Scholar

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B. T. Cheng and X. H. Tang, Infinitely many large energy solutions for Schrödinger-Kirchhoff type problem in $\mathbbR^N$,, \emph{J. Nonlinear Sci. Appl.}, 9 (2016), 652.   Google Scholar

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X. J. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian,, \emph{J. Differential Equations}, 256 (2014), 2965.  doi: 10.1016/j.jde.2014.01.027.  Google Scholar

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Y. X. Guo and J. J. Nie, Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger-Kirchhoff-type equations,, \emph{J. Math. Anal. Appl.}, 428 (2015), 1054.  doi: 10.1016/j.jmaa.2015.03.064.  Google Scholar

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X. M. He and W. M. Zou, Existen and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$,, \emph{J. Differential Equations}, 2 (2012), 1813.  doi: 10.1016/j.jde.2011.08.035.  Google Scholar

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J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 369 (2010), 564.  doi: 10.1016/j.jmaa.2010.03.059.  Google Scholar

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Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations,, \emph{J. Differential Equations}, 172 (2001), 257.  doi: 10.1006/jdeq.2000.3867.  Google Scholar

[17]

J. Liu, S. X. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 395 (2012), 608.  doi: 10.1016/j.jmaa.2012.05.063.  Google Scholar

[18]

H. L. Liu and H. B. Chen, Haibo Least energy nodal solution for quasilinear biharmonic equations with critical exponent in $\mathbbR^N$,, \emph{Appl. Math. Lett.}, 48 (2015), 85.  doi: 10.1016/j.aml.2015.03.002.  Google Scholar

[19]

T. F. Ma, Positive solutions for a nonlocal fourth order equation of Kirchhoff type,, \emph{Disc. Contin. Dyn. Syst.}, (2007), 694.   Google Scholar

[20]

T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings,, \emph{Appl. Math. Lett.}, 13 (2000), 11.  doi: 10.1016/S0893-9659(00)00026-4.  Google Scholar

[21]

T. F. Ma, Existence results and numerical solutions for a beam equation with nonlinear boundary conditions,, \emph{Appl. Numer. Math.}, 47 (2003), 189.  doi: 10.1016/S0168-9274(03)00065-5.  Google Scholar

[22]

A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,, \emph{Nonlinear Anal.}, 70 (2009), 1275.  doi: 10.1016/j.na.2008.02.011.  Google Scholar

[23]

P. J. McKenna and W. Walter, Traveling waves in a suspension bridge,, \emph{SIAM J. Appl. Math.}, 50 (1990), 703.  doi: 10.1137/0150041.  Google Scholar

[24]

J. J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations,, \emph{J. Math. Anal. Appl.}, 417 (2014), 65.  doi: 10.1016/j.jmaa.2014.03.027.  Google Scholar

[25]

J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential,, \emph{Nonlinear Anal.}, 75 (2012), 3470.  doi: 10.1016/j.na.2012.01.004.  Google Scholar

[26]

W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains,, \emph{J. Differential Equations}, 259 (2015), 1256.  doi: 10.1016/j.jde.2015.02.040.  Google Scholar

[27]

W. A. Strauss, Existence of solitary waves in higher dimensions,, \emph{Commun. Math. Phys.}, 55 (1977), 149.   Google Scholar

[28]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, \emph{Nonlinear Anal., 12 (2011), 1278.  doi: 10.1016/j.nonrwa.2010.09.023.  Google Scholar

[29]

F. L. Wang, M. Avci and Y. K. An, Existence of solutions for fourth-order elliptic equations of Kirchhoff type,, \emph{J. Math. Anal. Appl.}, 409 (2014), 140.  doi: 10.1016/j.jmaa.2013.07.003.  Google Scholar

[30]

M. Willem, Minimax Theorems,, Birkh\, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[31]

L. P. Xu and H. B. Chen, Nontrivial solutions for Kirchhoff-type problems with a parameter,, \emph{J. Math. Anal. Appl.}, 433 (2016), 455.  doi: 10.1016/j.jmaa.2015.07.035.  Google Scholar

[32]

L. P. Xu and H. B. Chen, Multiplicity results for fourth order elliptic equations of Kirchhoff-type,, \emph{Acta Math. Sci. Ser. B}, 35 (2015), 1067.  doi: 10.1016/S0252-9602(15)30040-0.  Google Scholar

[33]

L. P. Xu and H. B. Chen, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory,, \emph{Bound. Value Probl.}, 2014 (2014).  doi: 10.1186/s13661-014-0212-5.  Google Scholar

[34]

Y. L. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations,, \emph{J. Math. Anal. Appl.}, 375 (2011), 699.  doi: 10.1016/j.jmaa.2010.10.019.  Google Scholar

[35]

Y. W. Ye and C. L. Tang, Infinitely many solutions for fourth-order elliptic equations,, \emph{J. Math. Anal. Appl.}, 394 (2012), 841.  doi: 10.1016/j.jmaa.2012.04.041.  Google Scholar

[36]

Y. W. Ye and C. L. Tang, Existence and multiplicity of solutions for fourth-order elliptic equations in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 406 (2013), 335.  doi: 10.1016/j.jmaa.2013.04.079.  Google Scholar

[37]

Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, \emph{J. Math. Anal. Appl.}, 317 (2006), 456.  doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar

[38]

J. Zhang, X. H. Tang and W. Zhang, Existence of multiple solutions of Kirchhoff type equation with sign-changing potential,, \emph{Appl. Math. Compu.}, 242 (2014), 491.  doi: 10.1016/j.amc.2014.05.070.  Google Scholar

[39]

W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials,, \emph{J. Math. Anal. Appl.}, 407 (2013), 359.  doi: 10.1016/j.jmaa.2013.05.044.  Google Scholar

[40]

W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with sign-changing potential,, \emph{Taiwanese J. Math.}, 18 (2014), 645.  doi: 10.11650/tjm.18.2014.3584.  Google Scholar

[41]

W. Zhang, X. H. Tang and J. Zhang, Ground states for a class of asymptotically linear fourthorder elliptic equations,, \emph{Appl. Anal.}, 94 (2015), 2168.  doi: 10.1080/00036811.2014.979807.  Google Scholar

[42]

W. Zhang, X. H. Tang and J. Zhang, Existence and concentration of solutions for sublinear fourth-order elliptic equations,, \emph{Electron. J. Diff. Equ.}, 2015 (2015), 1.   Google Scholar

show all references

References:
[1]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 1153.  doi: 10.1007/s00033-013-0376-3.  Google Scholar

[2]

J. M. Ball, Initial-boundary value for an extensible beam,, \emph{J. Math. Anal. Appl.}, 42 (1973), 61.   Google Scholar

[3]

H. M. Berger, A new approach to the analysis of large deflections of plates,, \emph{J. Appl. Mech.}, 22 (1955), 465.   Google Scholar

[4]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$,, \emph{Commun. Partial Differ. Equ.}, 20 (1995), 1725.  doi: 10.1080/03605309508821149.  Google Scholar

[5]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology,, \emph{Ann. I. H. Poincar\'e Anal, 22 (2005), 259.  doi: 10.1016/j.anihpc.2004.07.005.  Google Scholar

[6]

T. Bartch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems,, \emph{J. Anal. Math.}, 96 (2005), 1.  doi: 10.1007/BF02787822.  Google Scholar

[7]

T. Bartsch, Z. Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations,, in \emph{Handbook of Differential Equations-Stationary Partial Differential Equations} (M. Chipot and P. Quittner eds.), (2005), 1.  doi: 10.1016/S1874-5733(05)80009-9.  Google Scholar

[8]

B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, \emph{Nonlinear Anal.}, 71 (2009), 4883.  doi: 10.1016/j.na.2009.03.065.  Google Scholar

[9]

B. T. Cheng, X. Wu and J. Liu, Multiple solutions for a class of Kirchhoff-type problems with concave nonlinearity,, \emph{NoDEA Nonl. Diff. Equa. Appl.}, 19 (2012), 521.  doi: 10.1007/s00030-011-0141-2.  Google Scholar

[10]

B. T. Cheng and X. H. Tang, Infinitely many large energy solutions for Schrödinger-Kirchhoff type problem in $\mathbbR^N$,, \emph{J. Nonlinear Sci. Appl.}, 9 (2016), 652.   Google Scholar

[11]

X. J. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian,, \emph{J. Differential Equations}, 256 (2014), 2965.  doi: 10.1016/j.jde.2014.01.027.  Google Scholar

[12]

E. N. Dancer and Y. H. Du, On sign-changing solutions of certain semilinear elliptic problems,, \emph{Appl. Anal.}, 56 (1995), 193.  doi: 10.1080/00036819508840321.  Google Scholar

[13]

Y. X. Guo and J. J. Nie, Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger-Kirchhoff-type equations,, \emph{J. Math. Anal. Appl.}, 428 (2015), 1054.  doi: 10.1016/j.jmaa.2015.03.064.  Google Scholar

[14]

X. M. He and W. M. Zou, Existen and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$,, \emph{J. Differential Equations}, 2 (2012), 1813.  doi: 10.1016/j.jde.2011.08.035.  Google Scholar

[15]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 369 (2010), 564.  doi: 10.1016/j.jmaa.2010.03.059.  Google Scholar

[16]

Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations,, \emph{J. Differential Equations}, 172 (2001), 257.  doi: 10.1006/jdeq.2000.3867.  Google Scholar

[17]

J. Liu, S. X. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 395 (2012), 608.  doi: 10.1016/j.jmaa.2012.05.063.  Google Scholar

[18]

H. L. Liu and H. B. Chen, Haibo Least energy nodal solution for quasilinear biharmonic equations with critical exponent in $\mathbbR^N$,, \emph{Appl. Math. Lett.}, 48 (2015), 85.  doi: 10.1016/j.aml.2015.03.002.  Google Scholar

[19]

T. F. Ma, Positive solutions for a nonlocal fourth order equation of Kirchhoff type,, \emph{Disc. Contin. Dyn. Syst.}, (2007), 694.   Google Scholar

[20]

T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings,, \emph{Appl. Math. Lett.}, 13 (2000), 11.  doi: 10.1016/S0893-9659(00)00026-4.  Google Scholar

[21]

T. F. Ma, Existence results and numerical solutions for a beam equation with nonlinear boundary conditions,, \emph{Appl. Numer. Math.}, 47 (2003), 189.  doi: 10.1016/S0168-9274(03)00065-5.  Google Scholar

[22]

A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,, \emph{Nonlinear Anal.}, 70 (2009), 1275.  doi: 10.1016/j.na.2008.02.011.  Google Scholar

[23]

P. J. McKenna and W. Walter, Traveling waves in a suspension bridge,, \emph{SIAM J. Appl. Math.}, 50 (1990), 703.  doi: 10.1137/0150041.  Google Scholar

[24]

J. J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations,, \emph{J. Math. Anal. Appl.}, 417 (2014), 65.  doi: 10.1016/j.jmaa.2014.03.027.  Google Scholar

[25]

J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential,, \emph{Nonlinear Anal.}, 75 (2012), 3470.  doi: 10.1016/j.na.2012.01.004.  Google Scholar

[26]

W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains,, \emph{J. Differential Equations}, 259 (2015), 1256.  doi: 10.1016/j.jde.2015.02.040.  Google Scholar

[27]

W. A. Strauss, Existence of solitary waves in higher dimensions,, \emph{Commun. Math. Phys.}, 55 (1977), 149.   Google Scholar

[28]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, \emph{Nonlinear Anal., 12 (2011), 1278.  doi: 10.1016/j.nonrwa.2010.09.023.  Google Scholar

[29]

F. L. Wang, M. Avci and Y. K. An, Existence of solutions for fourth-order elliptic equations of Kirchhoff type,, \emph{J. Math. Anal. Appl.}, 409 (2014), 140.  doi: 10.1016/j.jmaa.2013.07.003.  Google Scholar

[30]

M. Willem, Minimax Theorems,, Birkh\, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[31]

L. P. Xu and H. B. Chen, Nontrivial solutions for Kirchhoff-type problems with a parameter,, \emph{J. Math. Anal. Appl.}, 433 (2016), 455.  doi: 10.1016/j.jmaa.2015.07.035.  Google Scholar

[32]

L. P. Xu and H. B. Chen, Multiplicity results for fourth order elliptic equations of Kirchhoff-type,, \emph{Acta Math. Sci. Ser. B}, 35 (2015), 1067.  doi: 10.1016/S0252-9602(15)30040-0.  Google Scholar

[33]

L. P. Xu and H. B. Chen, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory,, \emph{Bound. Value Probl.}, 2014 (2014).  doi: 10.1186/s13661-014-0212-5.  Google Scholar

[34]

Y. L. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations,, \emph{J. Math. Anal. Appl.}, 375 (2011), 699.  doi: 10.1016/j.jmaa.2010.10.019.  Google Scholar

[35]

Y. W. Ye and C. L. Tang, Infinitely many solutions for fourth-order elliptic equations,, \emph{J. Math. Anal. Appl.}, 394 (2012), 841.  doi: 10.1016/j.jmaa.2012.04.041.  Google Scholar

[36]

Y. W. Ye and C. L. Tang, Existence and multiplicity of solutions for fourth-order elliptic equations in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 406 (2013), 335.  doi: 10.1016/j.jmaa.2013.04.079.  Google Scholar

[37]

Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, \emph{J. Math. Anal. Appl.}, 317 (2006), 456.  doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar

[38]

J. Zhang, X. H. Tang and W. Zhang, Existence of multiple solutions of Kirchhoff type equation with sign-changing potential,, \emph{Appl. Math. Compu.}, 242 (2014), 491.  doi: 10.1016/j.amc.2014.05.070.  Google Scholar

[39]

W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials,, \emph{J. Math. Anal. Appl.}, 407 (2013), 359.  doi: 10.1016/j.jmaa.2013.05.044.  Google Scholar

[40]

W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with sign-changing potential,, \emph{Taiwanese J. Math.}, 18 (2014), 645.  doi: 10.11650/tjm.18.2014.3584.  Google Scholar

[41]

W. Zhang, X. H. Tang and J. Zhang, Ground states for a class of asymptotically linear fourthorder elliptic equations,, \emph{Appl. Anal.}, 94 (2015), 2168.  doi: 10.1080/00036811.2014.979807.  Google Scholar

[42]

W. Zhang, X. H. Tang and J. Zhang, Existence and concentration of solutions for sublinear fourth-order elliptic equations,, \emph{Electron. J. Diff. Equ.}, 2015 (2015), 1.   Google Scholar

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