# American Institute of Mathematical Sciences

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

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

 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.
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

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##### 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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