American Institute of Mathematical Sciences

Advanced
August  2012, 5(4): 741-751. doi: 10.3934/dcdss.2012.5.741

Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator

Pasquale Candito 1, and  Giovanni Molica Bisci 2,

1. 

DIMET - Facoltà di Ingegneria, Università di Reggio Calabria, Località Feo di Vito, 89100 Reggio Calabria, Italy
2. 

Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di Vito, 89100 Reggio Calabria, Italy

Received  February 2011 Revised  April 2011 Published  November 2011

The existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the $p$--biharmonic operator is investigated. Our approach is chiefly based on critical point theory.
Keywords: Critical point theory., Three weak solutions, Navier boundary value problem, $p$--biharmonic type operators.
Mathematics Subject Classification: Primary: 35J40, 35J6.
Citation: Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741
References:
[1]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations,, J. Math. Anal. Appl., 343 (2008), 1166.  doi: 10.1016/j.jmaa.2008.01.049.  Google Scholar

[2]

G. Bonanno and B. Di Bella, A fourth-order boundary value problem for a Sturm-Liouville type equation,, Appl. Math. Comput., 217 (2010), 3635.  doi: 10.1016/j.amc.2010.10.019.  Google Scholar

[3]

G. Bonanno and B. Di Bella, Infinitely many solutions for a fourth-order elastic beam equation,, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 357.  doi: 10.1007/s00030-011-0099-0.  Google Scholar

[4]

G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities,, J. Differential Equations, 244 (2008), 3031.  doi: 10.1016/j.jde.2008.02.025.  Google Scholar

[5]

G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition,, Appl. Anal., 89 (2010), 1.  doi: 10.1080/00036810903397438.  Google Scholar

[6]

H. M. Guo and D. Geng, Infinitely many solutions for the Dirichlet problem involving the $p$-biharmonic-like equation,, J. South China Normal Univ. Natur. Sci. Ed., 2009 (2009), 18.   Google Scholar

[7]

M. R. Grossinho, L. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation,, Appl. Math. Lett., 18 (2005), 439.   Google Scholar

[8]

G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations,, Nonlinear Anal., 68 (2008), 3646.  doi: 10.1016/j.na.2007.04.007.  Google Scholar

[9]

C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the $p$-biharmonic,, Nonlinear Anal., 72 (2010), 1339.  doi: 10.1016/j.na.2009.08.011.  Google Scholar

[10]

X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters,, J. Math. Anal. Appl., 327 (2007), 362.  doi: 10.1016/j.jmaa.2006.04.021.  Google Scholar

[11]

A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem,, Nonlinear Anal., 31 (1998), 895.  doi: 10.1016/S0362-546X(97)00446-X.  Google Scholar

[12]

B. Ricceri, A general variational principle and some of its applications. Fixed point theory with applications in nonlinear analysis,, J. Comput. Appl. Math., 133 (2000), 401.  doi: 10.1016/S0377-0427(99)00269-1.  Google Scholar

[13]

G. Talenti, Elliptic equations and rearrangements,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697.   Google Scholar

[14]

Y. Wang and Y. Shen, Infinitely many sign-changing solutions for a class of biharmonic equation without symmetry,, Nonlinear Anal., 71 (2009), 967.  doi: 10.1016/j.na.2008.11.052.  Google Scholar

[15]

W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of $p$-biharmonic type,, J. Math. Anal. Appl., 348 (2008), 730.  doi: 10.1016/j.jmaa.2008.07.068.  Google Scholar

[16]

Z. Yang, D. Geng and H. Yan, Existence of nontrivial solutions in $p$-biharmonic problems with critical growth,, (Chinese) Chinese Ann. Math. Ser. A, 27 (2006), 129.   Google Scholar

show all references

References:
[1]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations,, J. Math. Anal. Appl., 343 (2008), 1166.  doi: 10.1016/j.jmaa.2008.01.049.  Google Scholar

[2]

G. Bonanno and B. Di Bella, A fourth-order boundary value problem for a Sturm-Liouville type equation,, Appl. Math. Comput., 217 (2010), 3635.  doi: 10.1016/j.amc.2010.10.019.  Google Scholar

[3]

G. Bonanno and B. Di Bella, Infinitely many solutions for a fourth-order elastic beam equation,, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 357.  doi: 10.1007/s00030-011-0099-0.  Google Scholar

[4]

G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities,, J. Differential Equations, 244 (2008), 3031.  doi: 10.1016/j.jde.2008.02.025.  Google Scholar

[5]

G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition,, Appl. Anal., 89 (2010), 1.  doi: 10.1080/00036810903397438.  Google Scholar

[6]

H. M. Guo and D. Geng, Infinitely many solutions for the Dirichlet problem involving the $p$-biharmonic-like equation,, J. South China Normal Univ. Natur. Sci. Ed., 2009 (2009), 18.   Google Scholar

[7]

M. R. Grossinho, L. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation,, Appl. Math. Lett., 18 (2005), 439.   Google Scholar

[8]

G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations,, Nonlinear Anal., 68 (2008), 3646.  doi: 10.1016/j.na.2007.04.007.  Google Scholar

[9]

C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the $p$-biharmonic,, Nonlinear Anal., 72 (2010), 1339.  doi: 10.1016/j.na.2009.08.011.  Google Scholar

[10]

X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters,, J. Math. Anal. Appl., 327 (2007), 362.  doi: 10.1016/j.jmaa.2006.04.021.  Google Scholar

[11]

A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem,, Nonlinear Anal., 31 (1998), 895.  doi: 10.1016/S0362-546X(97)00446-X.  Google Scholar

[12]

B. Ricceri, A general variational principle and some of its applications. Fixed point theory with applications in nonlinear analysis,, J. Comput. Appl. Math., 133 (2000), 401.  doi: 10.1016/S0377-0427(99)00269-1.  Google Scholar

[13]

G. Talenti, Elliptic equations and rearrangements,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697.   Google Scholar

[14]

Y. Wang and Y. Shen, Infinitely many sign-changing solutions for a class of biharmonic equation without symmetry,, Nonlinear Anal., 71 (2009), 967.  doi: 10.1016/j.na.2008.11.052.  Google Scholar

[15]

W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of $p$-biharmonic type,, J. Math. Anal. Appl., 348 (2008), 730.  doi: 10.1016/j.jmaa.2008.07.068.  Google Scholar

[16]

Z. Yang, D. Geng and H. Yan, Existence of nontrivial solutions in $p$-biharmonic problems with critical growth,, (Chinese) Chinese Ann. Math. Ser. A, 27 (2006), 129.   Google Scholar

[1]

John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337
[2]

John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269
[3]

Marta García-Huidobro, Raul Manásevich. A three point boundary value problem containing the operator. Conference Publications, 2003, 2003 (Special) : 313-319. doi: 10.3934/proc.2003.2003.313
[4]

Wenying Feng. Solutions and positive solutions for some three-point boundary value problems. Conference Publications, 2003, 2003 (Special) : 263-272. doi: 10.3934/proc.2003.2003.263
[5]

J. R. L. Webb. Remarks on positive solutions of some three point boundary value problems. Conference Publications, 2003, 2003 (Special) : 905-915. doi: 10.3934/proc.2003.2003.905
[6]

Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485
[7]

Zhongliang Wang. Nonradial positive solutions for a biharmonic critical growth problem. Communications on Pure & Applied Analysis, 2012, 11 (2) : 517-545. doi: 10.3934/cpaa.2012.11.517
[8]

Linh Nguyen, Irina Perfilieva, Michal Holčapek. Boundary value problem: Weak solutions induced by fuzzy partitions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 715-732. doi: 10.3934/dcdsb.2019263
[9]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[10]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[11]

Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775
[12]

K. Q. Lan. Properties of kernels and eigenvalues for three point boundary value problems. Conference Publications, 2005, 2005 (Special) : 546-555. doi: 10.3934/proc.2005.2005.546
[13]

Elvise Berchio, Filippo Gazzola. Positive solutions to a linearly perturbed critical growth biharmonic problem. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 809-823. doi: 10.3934/dcdss.2011.4.809
[14]

Erik Ekström, Johan Tysk. A boundary point lemma for Black-Scholes type operators. Communications on Pure & Applied Analysis, 2006, 5 (3) : 505-514. doi: 10.3934/cpaa.2006.5.505
[15]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[16]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[17]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834
[18]

Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148
[19]

Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759
[20]

John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences