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

Giuseppina Barletta; Roberto Livrea; Nikolaos S. Papageorgiou

doi:10.3934/cpaa.2014.13.1075

Article Contents

2014, Volume 13, Issue 3: 1075-1086. Doi: 10.3934/cpaa.2014.13.1075

This issue Previous Article Multi-valued solutions to a class of parabolic Monge-Ampère equations Next Article Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions

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

1.
Department DICEAM, University of Reggio Calabria, Reggio Calabria, 89100
2.
Department MECMAT, Engineering Faculty, University of Reggio Calabria, Reggio Calabria, 89100

Received: February 28, 2013

Revised: June 30, 2013

Published: April 2015

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 34B15, 34B18; Secondary: 34C25.

Citation:

Related Papers

Cited by

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.
[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.
[3]	S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nonlinear reasonant periodic problems with concave terms, J. Math Anal. Appl., 375 (2011), 342-364.
[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.
[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.
[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.
[7]	N. Dunford and J. Schwartz, Linear Operators I, Wiley-Interscience, New York, 1958.
[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.
[9]	L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.
[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.
[11]	E. Papageorgiou and N. S. Papageorgiou, Two nontrivial solutions for quasilinear periodic problems, Proceedings Amer. Math. Soc., 132 (2004), 429-434.
[12]	J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
[13]	X. Yang, Multiple periodic solutions of a class of $p$-Laplacian, J. Math. Anal. Appl., 314 (2006), 17-29.