• 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

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 & Applied Analysis, 2016, 15 (6) : 2161-2177. doi: 10.3934/cpaa.2016032
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]

J. Math. Anal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059.  Google Scholar

[16]

J. Differential Equations, 172 (2001), 257-299. doi: 10.1006/jdeq.2000.3867.  Google Scholar

[17]

J. Math. Anal. Appl., 395 (2012), 608-615. doi: 10.1016/j.jmaa.2012.05.063.  Google Scholar

[18]

Appl. Math. Lett., 48 (2015), 85-90. doi: 10.1016/j.aml.2015.03.002.  Google Scholar

[19]

Disc. Contin. Dyn. Syst., Supplement (2007), 694-703.  Google Scholar

[20]

Appl. Math. Lett., 13 (2000), 11-15. doi: 10.1016/S0893-9659(00)00026-4.  Google Scholar

[21]

Appl. Numer. Math., 47 (2003), 189-196. doi: 10.1016/S0168-9274(03)00065-5.  Google Scholar

[22]

Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011.  Google Scholar

[23]

SIAM J. Appl. Math., 50 (1990), 703-715. doi: 10.1137/0150041.  Google Scholar

[24]

J. Math. Anal. Appl., 417 (2014), 65-79. doi: 10.1016/j.jmaa.2014.03.027.  Google Scholar

[25]

Nonlinear Anal., 75 (2012), 3470-3479. doi: 10.1016/j.na.2012.01.004.  Google Scholar

[26]

J. Differential Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040.  Google Scholar

[27]

Commun. Math. Phys., 55 (1977), 149-162.  Google Scholar

[28]

Nonlinear Anal., Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.  Google Scholar

[29]

J. Math. Anal. Appl., 409 (2014), 140-146. doi: 10.1016/j.jmaa.2013.07.003.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

Bound. Value Probl., 2014 (2014), 212. doi: 10.1186/s13661-014-0212-5.  Google Scholar

[34]

J. Math. Anal. Appl., 375 (2011), 699-705. doi: 10.1016/j.jmaa.2010.10.019.  Google Scholar

[35]

J. Math. Anal. Appl., 394 (2012), 841-854. doi: 10.1016/j.jmaa.2012.04.041.  Google Scholar

[36]

J. Math. Anal. Appl., 406 (2013), 335-351. doi: 10.1016/j.jmaa.2013.04.079.  Google Scholar

[37]

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

[38]

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

[39]

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

[41]

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

[42]

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

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]

J. Math. Anal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059.  Google Scholar

[16]

J. Differential Equations, 172 (2001), 257-299. doi: 10.1006/jdeq.2000.3867.  Google Scholar

[17]

J. Math. Anal. Appl., 395 (2012), 608-615. doi: 10.1016/j.jmaa.2012.05.063.  Google Scholar

[18]

Appl. Math. Lett., 48 (2015), 85-90. doi: 10.1016/j.aml.2015.03.002.  Google Scholar

[19]

Disc. Contin. Dyn. Syst., Supplement (2007), 694-703.  Google Scholar

[20]

Appl. Math. Lett., 13 (2000), 11-15. doi: 10.1016/S0893-9659(00)00026-4.  Google Scholar

[21]

Appl. Numer. Math., 47 (2003), 189-196. doi: 10.1016/S0168-9274(03)00065-5.  Google Scholar

[22]

Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011.  Google Scholar

[23]

SIAM J. Appl. Math., 50 (1990), 703-715. doi: 10.1137/0150041.  Google Scholar

[24]

J. Math. Anal. Appl., 417 (2014), 65-79. doi: 10.1016/j.jmaa.2014.03.027.  Google Scholar

[25]

Nonlinear Anal., 75 (2012), 3470-3479. doi: 10.1016/j.na.2012.01.004.  Google Scholar

[26]

J. Differential Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040.  Google Scholar

[27]

Commun. Math. Phys., 55 (1977), 149-162.  Google Scholar

[28]

Nonlinear Anal., Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.  Google Scholar

[29]

J. Math. Anal. Appl., 409 (2014), 140-146. doi: 10.1016/j.jmaa.2013.07.003.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

Bound. Value Probl., 2014 (2014), 212. doi: 10.1186/s13661-014-0212-5.  Google Scholar

[34]

J. Math. Anal. Appl., 375 (2011), 699-705. doi: 10.1016/j.jmaa.2010.10.019.  Google Scholar

[35]

J. Math. Anal. Appl., 394 (2012), 841-854. doi: 10.1016/j.jmaa.2012.04.041.  Google Scholar

[36]

J. Math. Anal. Appl., 406 (2013), 335-351. doi: 10.1016/j.jmaa.2013.04.079.  Google Scholar

[37]

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

[38]

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

[39]

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

[41]

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

[42]

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

[1]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454

[2]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091

[3]

Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021115

[4]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[5]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027

[6]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[7]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234

[8]

Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021058

[9]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[10]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019

[11]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016

[12]

Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021069

[13]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[14]

Tôn Việt Tạ. Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021050

[15]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060

[16]

Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021129

[17]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[18]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[19]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083

[20]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (123)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]