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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), 1166-1176. 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), 3635-3640. 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), 357-368. 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), 3031-3059. 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), 1-10. 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, 18-21, 28.  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), 439-444.  Google Scholar

[8]

G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear Anal., 68 (2008), 3646-3656. 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), 1339-1347. 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), 362-375. 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), 895-908. 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), 401-410. 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), 697-718.  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), 967-977. 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), 730-738. 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), 129-142.  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), 1166-1176. 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), 3635-3640. 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), 357-368. 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), 3031-3059. 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), 1-10. 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, 18-21, 28.  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), 439-444.  Google Scholar

[8]

G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear Anal., 68 (2008), 3646-3656. 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), 1339-1347. 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), 362-375. 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), 895-908. 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), 401-410. 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), 697-718.  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), 967-977. 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), 730-738. 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), 129-142.  Google Scholar

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