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

Wen  Zhang; Xianhua Tang; Bitao  Cheng; Jian  Zhang

doi:10.3934/cpaa.2016032

Previous Article
Sharp well-posedness for the Chen-Lee equation
CPAA Home
This Issue
Next Article
Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates

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

Abstract
Related Papers

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, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3. Google Scholar
[2]	J. M. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90. Google Scholar
[3]	H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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, J. Differential Equations, 172 (2001), 257-299. 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 $\mathbbR^N$, J. Math. Anal. Appl., 395 (2012), 608-615. 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 $\mathbbR^N$, Appl. Math. Lett., 48 (2015), 85-90. 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, Disc. Contin. Dyn. Syst., Supplement (2007), 694-703. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	M. Willem, Minimax Theorems, Birkhäuser, Berlin, 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, J. Math. Anal. Appl., 433 (2016), 455-472. 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, Acta Math. Sci. Ser. B, 35 (2015), 1067-1076. 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, Bound. Value Probl., 2014 (2014), 212. 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, J. Math. Anal. Appl., 375 (2011), 699-705. 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, J. Math. Anal. Appl., 394 (2012), 841-854. 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 $\mathbbR^N$, J. Math. Anal. Appl., 406 (2013), 335-351. 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, J. Math. Anal. Appl., 317 (2006), 456-463. 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, Appl. Math. Compu., 242 (2014), 491-499. 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, J. Math. Anal. Appl., 407 (2013), 359-368. 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, Taiwanese J. Math., 18 (2014), 645-659. 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, Appl. Anal., 94 (2015), 2168-2174. 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, Electron. J. Diff. Equ., 2015 (2015), 1-9. 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, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3. Google Scholar
[2]	J. M. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90. Google Scholar
[3]	H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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, J. Differential Equations, 172 (2001), 257-299. 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 $\mathbbR^N$, J. Math. Anal. Appl., 395 (2012), 608-615. 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 $\mathbbR^N$, Appl. Math. Lett., 48 (2015), 85-90. 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, Disc. Contin. Dyn. Syst., Supplement (2007), 694-703. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	M. Willem, Minimax Theorems, Birkhäuser, Berlin, 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, J. Math. Anal. Appl., 433 (2016), 455-472. 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, Acta Math. Sci. Ser. B, 35 (2015), 1067-1076. 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, Bound. Value Probl., 2014 (2014), 212. 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, J. Math. Anal. Appl., 375 (2011), 699-705. 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, J. Math. Anal. Appl., 394 (2012), 841-854. 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 $\mathbbR^N$, J. Math. Anal. Appl., 406 (2013), 335-351. 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, J. Math. Anal. Appl., 317 (2006), 456-463. 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, Appl. Math. Compu., 242 (2014), 491-499. 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, J. Math. Anal. Appl., 407 (2013), 359-368. 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, Taiwanese J. Math., 18 (2014), 645-659. 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, Appl. Anal., 94 (2015), 2168-2174. 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, Electron. J. Diff. Equ., 2015 (2015), 1-9. Google Scholar

[1]	Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137
[2]	Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883
[3]	Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055
[4]	Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256
[5]	Jie Yang, Haibo Chen. Multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3065-3092. doi: 10.3934/cpaa.2021096
[6]	Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389
[7]	To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694-703. doi: 10.3934/proc.2007.2007.694
[8]	Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151
[9]	Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737
[10]	Daniela Giachetti, Francesco Petitta, Sergio Segura de León. Elliptic equations having a singular quadratic gradient term and a changing sign datum. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1875-1895. doi: 10.3934/cpaa.2012.11.1875
[11]	Gabriele Cora, Alessandro Iacopetti. Sign-changing bubble-tower solutions to fractional semilinear elliptic problems. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6149-6173. doi: 10.3934/dcds.2019268
[12]	Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
[13]	Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565
[14]	Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383
[15]	Zhiqing Liu, Zhong Bo Fang. Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term. Communications on Pure & Applied Analysis, 2020, 19 (2) : 941-966. doi: 10.3934/cpaa.2020043
[16]	J. Húska, Peter Poláčik, M.V. Safonov. Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations. Conference Publications, 2005, 2005 (Special) : 427-435. doi: 10.3934/proc.2005.2005.427
[17]	Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917
[18]	Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure & Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004
[19]	Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883
[20]	M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057

2020 Impact Factor: 1.916

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors