# 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:
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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,, \emph{J. Differential Equation}, 243 (2007), 504.   Google Scholar [2] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neuamann problems,, \emph{Ann. Mat. Pura Appl.}, 186 (2009), 679.   Google Scholar [3] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nonlinear reasonant periodic problems with concave terms,, \emph{J. Math Anal. Appl.}, 375 (2011), 342.   Google Scholar [4] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Positive solutions for nonlinear periodic problems with cnaceve terms,, \emph{J. Math Anal. Appl.}, 381 (2011), 866.   Google Scholar [5] M. Del Pino, R. Manasevich and A. Murua, Exixtence and multiplcity of solutions with prescribed period for second order quasilinear ODE,, \emph{Nonlinear Anal.}, 18 (1992), 79.   Google Scholar [6] P. Drabek and R. Manasevich, On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian,, \emph{Differential Integral Equations}, 12 (1999), 773.   Google Scholar [7] N. Dunford and J. Schwartz, Linear Operators I,, Wiley-Interscience, (1958).   Google Scholar [8] L. Gasinski, Positive solutions for reasonant boundary value problems with the scalar $p$-Laplacian and a nonsmooth potential,, \emph{Discrete Cont. Dynam. Systems}, 17 (2007), 143.   Google Scholar [9] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis,, Chapman & Hall/CRC, (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,, \emph{Commun. Pure Appl. Anal.}, 8 (2009), 1421.   Google Scholar [11] E. Papageorgiou and N. S. Papageorgiou, Two nontrivial solutions for quasilinear periodic problems,, \emph{Proceedings Amer. Math. Soc.}, 132 (2004), 429.   Google Scholar [12] J. Vazquez, A strong maximum principle for some quasilinear elliptic equations,, \emph{Appl. Math. Optim.}, 12 (1984), 191.   Google Scholar [13] X. Yang, Multiple periodic solutions of a class of $p$-Laplacian,, \emph{J. Math. Anal. Appl.}, 314 (2006), 17.   Google Scholar
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