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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
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show all references

References:
[1]

Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3.  Google Scholar

[2]

J. Math. Anal. Appl., 42 (1973), 61-90.  Google Scholar

[3]

J. Appl. Mech., 22 (1955), 465-472.  Google Scholar

[4]

Commun. Partial Differ. Equ., 20 (1995), 1725-1741. doi: 10.1080/03605309508821149.  Google Scholar

[5]

Ann. I. H. Poincaré Anal, Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005.  Google Scholar

[6]

J. Anal. Math., 96 (2005), 1-18. doi: 10.1007/BF02787822.  Google Scholar

[7]

in Handbook of Differential Equations-Stationary Partial Differential Equations (M. Chipot and P. Quittner eds.), vol. 2, Elsevier, 2005, pp. 1-55, Chapter 1. doi: 10.1016/S1874-5733(05)80009-9.  Google Scholar

[8]

Nonlinear Anal., 71 (2009), 4883-4892. doi: 10.1016/j.na.2009.03.065.  Google Scholar

[9]

NoDEA Nonl. Diff. Equa. Appl., 19 (2012), 521-537. doi: 10.1007/s00030-011-0141-2.  Google Scholar

[10]

J. Nonlinear Sci. Appl., 9 (2016), 652-660.  Google Scholar

[11]

J. Differential Equations, 256 (2014), 2965-2992. doi: 10.1016/j.jde.2014.01.027.  Google Scholar

[12]

Appl. Anal., 56 (1995), 193-206. doi: 10.1080/00036819508840321.  Google Scholar

[13]

J. Math. Anal. Appl., 428 (2015), 1054-1069. doi: 10.1016/j.jmaa.2015.03.064.  Google Scholar

[14]

J. Differential Equations, 2 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035.  Google Scholar

[15]

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[16]

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[17]

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Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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J. Math. Anal. Appl., 433 (2016), 455-472. doi: 10.1016/j.jmaa.2015.07.035.  Google Scholar

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Acta Math. Sci. Ser. B, 35 (2015), 1067-1076. doi: 10.1016/S0252-9602(15)30040-0.  Google Scholar

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J. Math. Anal. Appl., 317 (2006), 456-463. doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar

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Appl. Math. Compu., 242 (2014), 491-499. doi: 10.1016/j.amc.2014.05.070.  Google Scholar

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J. Math. Anal. Appl., 407 (2013), 359-368. doi: 10.1016/j.jmaa.2013.05.044.  Google Scholar

[40]

Taiwanese J. Math., 18 (2014), 645-659. doi: 10.11650/tjm.18.2014.3584.  Google Scholar

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Appl. Anal., 94 (2015), 2168-2174. doi: 10.1080/00036811.2014.979807.  Google Scholar

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Electron. J. Diff. Equ., 2015 (2015), 1-9. Google Scholar

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