American Institute of Mathematical Sciences

Advanced
May  2014, 13(3): 1075-1086. doi: 10.3934/cpaa.2014.13.1075

A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian

Giuseppina Barletta 1, , Roberto Livrea 2, and  Nikolaos S. Papageorgiou 1,

1. 

Department DICEAM, University of Reggio Calabria, Reggio Calabria, 89100, Italy, Italy
2. 

Department MECMAT, Engineering Faculty, University of Reggio Calabria, Reggio Calabria, 89100

Received  March 2013 Revised  July 2013 Published  December 2013

We study a parametric nonlinear periodic problem driven by the scalar $p$-Laplacian. We show that if $\hat \lambda_1 >0$ is the first eigenvalue of the periodic scalar $p$-Laplacian and $\lambda> \hat \lambda_1$, then the problem has at least three nontrivial solutions one positive, one negative and the third nodal. Our approach is variational together with suitable truncation, perturbation and comparison techniques.
Keywords: parametric equation, second deformation theorem, extremal solutions., Constant sign and nodal solutions.
Mathematics Subject Classification: Primary: 34B15, 34B18; Secondary: 34C2.
Citation: Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Multiple nontrivial solutions for nonlinear periodic problems with the $p$-Laplacian, J. Differential Equation, 243 (2007), 504-535.  Google Scholar

[2]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neuamann problems, Ann. Mat. Pura Appl., 186 (2009), 679-719.  Google Scholar

[3]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nonlinear reasonant periodic problems with concave terms, J. Math Anal. Appl., 375 (2011), 342-364.  Google Scholar

[4]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Positive solutions for nonlinear periodic problems with cnaceve terms, J. Math Anal. Appl., 381 (2011), 866-883.  Google Scholar

[5]

M. Del Pino, R. Manasevich and A. Murua, Exixtence and multiplcity of solutions with prescribed period for second order quasilinear ODE, Nonlinear Anal., 18 (1992), 79-92.  Google Scholar

[6]

P. Drabek and R. Manasevich, On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian, Differential Integral Equations, 12 (1999), 773-788.  Google Scholar

[7]

N. Dunford and J. Schwartz, Linear Operators I, Wiley-Interscience, New York, 1958.  Google Scholar

[8]

L. Gasinski, Positive solutions for reasonant boundary value problems with the scalar $p$-Laplacian and a nonsmooth potential, Discrete Cont. Dynam. Systems, 17 (2007), 143-158.  Google Scholar

[9]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar

[10]

L. Gasinski and N. S. Papageorgiou, Three nontrivial soluions for periodic problems with the $p$-Laplacian and a $p$-superlinear nonlinearity, Commun. Pure Appl. Anal., 8 (2009), 1421-1437.  Google Scholar

[11]

E. Papageorgiou and N. S. Papageorgiou, Two nontrivial solutions for quasilinear periodic problems, Proceedings Amer. Math. Soc., 132 (2004), 429-434.  Google Scholar

[12]

J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  Google Scholar

[13]

X. Yang, Multiple periodic solutions of a class of $p$-Laplacian, J. Math. Anal. Appl., 314 (2006), 17-29.  Google Scholar

show all references

References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Multiple nontrivial solutions for nonlinear periodic problems with the $p$-Laplacian, J. Differential Equation, 243 (2007), 504-535.  Google Scholar

[2]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neuamann problems, Ann. Mat. Pura Appl., 186 (2009), 679-719.  Google Scholar

[3]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nonlinear reasonant periodic problems with concave terms, J. Math Anal. Appl., 375 (2011), 342-364.  Google Scholar

[4]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Positive solutions for nonlinear periodic problems with cnaceve terms, J. Math Anal. Appl., 381 (2011), 866-883.  Google Scholar

[5]

M. Del Pino, R. Manasevich and A. Murua, Exixtence and multiplcity of solutions with prescribed period for second order quasilinear ODE, Nonlinear Anal., 18 (1992), 79-92.  Google Scholar

[6]

P. Drabek and R. Manasevich, On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian, Differential Integral Equations, 12 (1999), 773-788.  Google Scholar

[7]

N. Dunford and J. Schwartz, Linear Operators I, Wiley-Interscience, New York, 1958.  Google Scholar

[8]

L. Gasinski, Positive solutions for reasonant boundary value problems with the scalar $p$-Laplacian and a nonsmooth potential, Discrete Cont. Dynam. Systems, 17 (2007), 143-158.  Google Scholar

[9]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar

[10]

L. Gasinski and N. S. Papageorgiou, Three nontrivial soluions for periodic problems with the $p$-Laplacian and a $p$-superlinear nonlinearity, Commun. Pure Appl. Anal., 8 (2009), 1421-1437.  Google Scholar

[11]

E. Papageorgiou and N. S. Papageorgiou, Two nontrivial solutions for quasilinear periodic problems, Proceedings Amer. Math. Soc., 132 (2004), 429-434.  Google Scholar

[12]

J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  Google Scholar

[13]

X. Yang, Multiple periodic solutions of a class of $p$-Laplacian, J. Math. Anal. Appl., 314 (2006), 17-29.  Google Scholar

[1]

Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6133-6166. doi: 10.3934/dcds.2016068
[2]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[3]

Gustavo S. Costa, Giovany M. Figueiredo. Existence and concentration of nodal solutions for a subcritical p&q equation. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5077-5095. doi: 10.3934/cpaa.2020227
[4]

Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326
[5]

Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095
[6]

Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561
[7]

Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405
[8]

A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253
[9]

Addolorata Salvatore. Sign--changing solutions for an asymptotically linear Schrödinger equation. Conference Publications, 2009, 2009 (Special) : 669-677. doi: 10.3934/proc.2009.2009.669
[10]

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
[11]

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
[12]

Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023
[13]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
[14]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2021038
[15]

Monica Lazzo, Paul G. Schmidt. Nodal properties of radial solutions for a class of polyharmonic equations. Conference Publications, 2007, 2007 (Special) : 634-643. doi: 10.3934/proc.2007.2007.634
[16]

Lei Zhang, Anfu Zhu, Aiguo Wu, Lingling Lv. Parametric solutions to the regulator-conjugate matrix equations. Journal of Industrial & Management Optimization, 2017, 13 (2) : 623-631. doi: 10.3934/jimo.2016036
[17]

Marianne Beringhier, Adrien Leygue, Francisco Chinesta. Parametric nonlinear PDEs with multiple solutions: A PGD approach. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 383-392. doi: 10.3934/dcdss.2016002
[18]

Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002
[19]

Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 25-36. doi: 10.3934/jimo.2018138
[20]

Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences