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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

 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 $b\rightarrow 0$.
Citation: Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Communications on Pure and 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, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3. [2] J. M. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90. [3] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. [4] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. [5] T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. I. H. Poincaré Anal, Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005. [6] T. Bartch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. doi: 10.1007/BF02787822. [7] T. Bartsch, Z. Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, 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. [8] B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892. doi: 10.1016/j.na.2009.03.065. [9] B. T. Cheng, X. Wu and J. Liu, Multiple solutions for a class of Kirchhoff-type problems with concave nonlinearity, NoDEA Nonl. Diff. Equa. Appl., 19 (2012), 521-537. doi: 10.1007/s00030-011-0141-2. [10] B. T. Cheng and X. H. Tang, Infinitely many large energy solutions for Schrödinger-Kirchhoff type problem in $\mathbbR^N$, J. Nonlinear Sci. Appl., 9 (2016), 652-660. [11] X. J. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992. doi: 10.1016/j.jde.2014.01.027. [12] E. N. Dancer and Y. H. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal., 56 (1995), 193-206. doi: 10.1080/00036819508840321. [13] Y. X. Guo and J. J. Nie, Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 428 (2015), 1054-1069. doi: 10.1016/j.jmaa.2015.03.064. [14] X. M. He and W. M. Zou, Existen and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$, J. Differential Equations, 2 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. [15] J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Anal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059. [16] Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299. doi: 10.1006/jdeq.2000.3867. [17] J. Liu, S. X. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbbR^N$, J. Math. Anal. Appl., 395 (2012), 608-615. doi: 10.1016/j.jmaa.2012.05.063. [18] H. L. Liu and H. B. Chen, Haibo Least energy nodal solution for quasilinear biharmonic equations with critical exponent in $\mathbbR^N$, Appl. Math. Lett., 48 (2015), 85-90. doi: 10.1016/j.aml.2015.03.002. [19] T. F. Ma, Positive solutions for a nonlocal fourth order equation of Kirchhoff type, Disc. Contin. Dyn. Syst., Supplement (2007), 694-703. [20] T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings, Appl. Math. Lett., 13 (2000), 11-15. doi: 10.1016/S0893-9659(00)00026-4. [21] T. F. Ma, Existence results and numerical solutions for a beam equation with nonlinear boundary conditions, Appl. Numer. Math., 47 (2003), 189-196. doi: 10.1016/S0168-9274(03)00065-5. [22] A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011. [23] P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715. doi: 10.1137/0150041. [24] J. J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 417 (2014), 65-79. doi: 10.1016/j.jmaa.2014.03.027. [25] J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479. doi: 10.1016/j.na.2012.01.004. [26] W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040. [27] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. [28] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$, Nonlinear Anal., Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023. [29] F. L. Wang, M. Avci and Y. K. An, Existence of solutions for fourth-order elliptic equations of Kirchhoff type, J. Math. Anal. Appl., 409 (2014), 140-146. doi: 10.1016/j.jmaa.2013.07.003. [30] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. [31] L. P. Xu and H. B. Chen, Nontrivial solutions for Kirchhoff-type problems with a parameter, J. Math. Anal. Appl., 433 (2016), 455-472. doi: 10.1016/j.jmaa.2015.07.035. [32] L. P. Xu and H. B. Chen, Multiplicity results for fourth order elliptic equations of Kirchhoff-type, Acta Math. Sci. Ser. B, 35 (2015), 1067-1076. doi: 10.1016/S0252-9602(15)30040-0. [33] L. P. Xu and H. B. Chen, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory, Bound. Value Probl., 2014 (2014), 212. doi: 10.1186/s13661-014-0212-5. [34] Y. L. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 375 (2011), 699-705. doi: 10.1016/j.jmaa.2010.10.019. [35] Y. W. Ye and C. L. Tang, Infinitely many solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 394 (2012), 841-854. doi: 10.1016/j.jmaa.2012.04.041. [36] Y. W. Ye and C. L. Tang, Existence and multiplicity of solutions for fourth-order elliptic equations in $\mathbbR^N$, J. Math. Anal. Appl., 406 (2013), 335-351. doi: 10.1016/j.jmaa.2013.04.079. [37] Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463. doi: 10.1016/j.jmaa.2005.06.102. [38] J. Zhang, X. H. Tang and W. Zhang, Existence of multiple solutions of Kirchhoff type equation with sign-changing potential, Appl. Math. Compu., 242 (2014), 491-499. doi: 10.1016/j.amc.2014.05.070. [39] W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials, J. Math. Anal. Appl., 407 (2013), 359-368. doi: 10.1016/j.jmaa.2013.05.044. [40] W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with sign-changing potential, Taiwanese J. Math., 18 (2014), 645-659. doi: 10.11650/tjm.18.2014.3584. [41] W. Zhang, X. H. Tang and J. Zhang, Ground states for a class of asymptotically linear fourthorder elliptic equations, Appl. Anal., 94 (2015), 2168-2174. doi: 10.1080/00036811.2014.979807. [42] W. Zhang, X. H. Tang and J. Zhang, Existence and concentration of solutions for sublinear fourth-order elliptic equations, Electron. J. Diff. Equ., 2015 (2015), 1-9.

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, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3. [2] J. M. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90. [3] H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. [4] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. [5] T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. I. H. Poincaré Anal, Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005. [6] T. Bartch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. doi: 10.1007/BF02787822. [7] T. Bartsch, Z. Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, 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. [8] B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892. doi: 10.1016/j.na.2009.03.065. [9] B. T. Cheng, X. Wu and J. Liu, Multiple solutions for a class of Kirchhoff-type problems with concave nonlinearity, NoDEA Nonl. Diff. Equa. Appl., 19 (2012), 521-537. doi: 10.1007/s00030-011-0141-2. [10] B. T. Cheng and X. H. Tang, Infinitely many large energy solutions for Schrödinger-Kirchhoff type problem in $\mathbbR^N$, J. Nonlinear Sci. Appl., 9 (2016), 652-660. [11] X. J. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992. doi: 10.1016/j.jde.2014.01.027. [12] E. N. Dancer and Y. H. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal., 56 (1995), 193-206. doi: 10.1080/00036819508840321. [13] Y. X. Guo and J. J. Nie, Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 428 (2015), 1054-1069. doi: 10.1016/j.jmaa.2015.03.064. [14] X. M. He and W. M. Zou, Existen and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^{3}$, J. Differential Equations, 2 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. [15] J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Anal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059. [16] Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299. doi: 10.1006/jdeq.2000.3867. [17] J. Liu, S. X. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbbR^N$, J. Math. Anal. Appl., 395 (2012), 608-615. doi: 10.1016/j.jmaa.2012.05.063. [18] H. L. Liu and H. B. Chen, Haibo Least energy nodal solution for quasilinear biharmonic equations with critical exponent in $\mathbbR^N$, Appl. Math. Lett., 48 (2015), 85-90. doi: 10.1016/j.aml.2015.03.002. [19] T. F. Ma, Positive solutions for a nonlocal fourth order equation of Kirchhoff type, Disc. Contin. Dyn. Syst., Supplement (2007), 694-703. [20] T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings, Appl. Math. Lett., 13 (2000), 11-15. doi: 10.1016/S0893-9659(00)00026-4. [21] T. F. Ma, Existence results and numerical solutions for a beam equation with nonlinear boundary conditions, Appl. Numer. Math., 47 (2003), 189-196. doi: 10.1016/S0168-9274(03)00065-5. [22] A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011. [23] P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), 703-715. doi: 10.1137/0150041. [24] J. J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations, J. Math. Anal. Appl., 417 (2014), 65-79. doi: 10.1016/j.jmaa.2014.03.027. [25] J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479. doi: 10.1016/j.na.2012.01.004. [26] W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040. [27] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. [28] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$, Nonlinear Anal., Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023. [29] F. L. Wang, M. Avci and Y. K. An, Existence of solutions for fourth-order elliptic equations of Kirchhoff type, J. Math. Anal. Appl., 409 (2014), 140-146. doi: 10.1016/j.jmaa.2013.07.003. [30] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. [31] L. P. Xu and H. B. Chen, Nontrivial solutions for Kirchhoff-type problems with a parameter, J. Math. Anal. Appl., 433 (2016), 455-472. doi: 10.1016/j.jmaa.2015.07.035. [32] L. P. Xu and H. B. Chen, Multiplicity results for fourth order elliptic equations of Kirchhoff-type, Acta Math. Sci. Ser. B, 35 (2015), 1067-1076. doi: 10.1016/S0252-9602(15)30040-0. [33] L. P. Xu and H. B. Chen, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory, Bound. Value Probl., 2014 (2014), 212. doi: 10.1186/s13661-014-0212-5. [34] Y. L. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 375 (2011), 699-705. doi: 10.1016/j.jmaa.2010.10.019. [35] Y. W. Ye and C. L. Tang, Infinitely many solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 394 (2012), 841-854. doi: 10.1016/j.jmaa.2012.04.041. [36] Y. W. Ye and C. L. Tang, Existence and multiplicity of solutions for fourth-order elliptic equations in $\mathbbR^N$, J. Math. Anal. Appl., 406 (2013), 335-351. doi: 10.1016/j.jmaa.2013.04.079. [37] Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463. doi: 10.1016/j.jmaa.2005.06.102. [38] J. Zhang, X. H. Tang and W. Zhang, Existence of multiple solutions of Kirchhoff type equation with sign-changing potential, Appl. Math. Compu., 242 (2014), 491-499. doi: 10.1016/j.amc.2014.05.070. [39] W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with general potentials, J. Math. Anal. Appl., 407 (2013), 359-368. doi: 10.1016/j.jmaa.2013.05.044. [40] W. Zhang, X. H. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with sign-changing potential, Taiwanese J. Math., 18 (2014), 645-659. doi: 10.11650/tjm.18.2014.3584. [41] W. Zhang, X. H. Tang and J. Zhang, Ground states for a class of asymptotically linear fourthorder elliptic equations, Appl. Anal., 94 (2015), 2168-2174. doi: 10.1080/00036811.2014.979807. [42] W. Zhang, X. H. Tang and J. Zhang, Existence and concentration of solutions for sublinear fourth-order elliptic equations, Electron. J. Diff. Equ., 2015 (2015), 1-9.
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