Abstract
We study the existence of positive solutions for a nonlinear periodic problem driven by the scalar p-Laplacian and having a nonsmooth potential. We impose a nonuniform nonresonance condition at + ∞ and a uniform nonresonance condition at 0 + . Using degree theoretic argument based on a fixed point index for multifunctions, we prove the existence of a strict positive solution.
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Agarwal, R.P., Lü, H., O’Regan, D.: Eigenvalues and the one-dimensional p-Laplacian. J. Math. Anal. Appl. 266, 383–400 (2002)
Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976)
Bader, R.: A topological fixed-point index theory for evolution inclusions. Z. Anal. Anwendungen 20, 3–15 (2001)
Ben Naoum, A.K., De Coster, C.: On the existence and multiplicity of positive solutions of the p-Laplacian separated boundary value problem. Differential Integral Equations 10, 1093–1112 (1997)
De Coster, C.: Pairs of positive solutions for the one-dimensional p-Laplacian. Nonlinear Anal. 23, 669–681 (1994)
Gasiński, L., Papageorgiou, N.S.: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman and Hall/CRC Press, Boca Raton (2005)
Manásevich, R.F., Njoku, F., Zanolin, F.: Positive solutions for the one dimensional p-Laplacian. Differential Integral Equations 8, 213–222 (1995)
Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)
Vázquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984)
Wang, J.Y.: On the existence of positive solutions for the one-dimensional p-Laplacian. Proc. Amer. Math. Soc. 125, 2275–2283 (1997)
Zhang, M.: The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials. J. London Math. Soc. 64, 125–143 (2001)
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This paper has been partially supported by the State Committee for Scientific Research of Poland (KBN) under research grants nr. 2 P03A 003 25 and nr. 4 T07A 027 26.
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Denkowski, Z., Gasiński, L. & Papageorgiou, N.S. Positive Solutions for Nonlinear Periodic Problems with the Scalar p-Laplacian. Set-Valued Anal 16, 539–561 (2008). https://doi.org/10.1007/s11228-007-0059-3
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DOI: https://doi.org/10.1007/s11228-007-0059-3
Keywords
- Nonsmooth potential
- Scalar p-Laplacian
- Generalized subdifferential
- Weighted eigenvalue problem
- Nonuniform nonresonance
- Fixed point index