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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

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