Abstract
In this paper, we study the existence of multiple solutions for nonlinear scalar periodic problems at resonance with p-Laplacian-like operator. Using the Ekeland variational principle a two-solution theorem is obtained and using also a local linking theorem a three-solution theorem is proved.
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References
Adly S., Motreanu, D. (2000) Periodic solutions for second-order differential equations involving nonconvex superpotentials. J. Global Optim. 17:9–17
Bartolo P., Benci V., Fortunato D. (1983) Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7:981–1012
Cerami G. (1978) Un criterio di esistenza per i punti critici su varietá illimitate. Rend. Accad. Sci. Let. Ist. Lombardo 112:332–336
Chang K.-C. (1981) Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80:102–129
Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983).
Dang H., Oppenheimer S.F. (1996) Existence and uniqueness results for some nonlinear boundary value problems. J. Math. Anal. Appl. 198:35–48
del Pino M.A., Manásevich R.F., Murúa A.E. (1992) Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE. Nonlinear Anal. 18:79–92
Denkowski, Z., Gasiński, L., Papageorgiou, N.S.: Existence of positive and of multiple solutions for nonlinear periodic problems. Nonlinear Anal. to appear.
Fabry C., Fayyad D. (1992) Periodic solutions of second order differential equations with a p-Laplacian and asymmetric nonlinearities. Rend. Ist. Mat. Univ. Trieste. 24:207–227
Gasiński L., Papageorgiou N.S. (2002) A Multiplicity result for nonlinear second order periodic equations with nonsmooth potential. Bull. Belg. Math. Soc. 9:245–258
Gasiński L., Papageorgiou N.S. (2003) On the existence of multiple periodic solutions for equations driven by the p-Laplacian and with a non-smooth potential. Proc. Edinburgh Math. Soc. 46:229–249
Gasiński L., Papageorgiou N.S.: Nonsmooth critical Point Theory and Nonlinear Boundary Value Problems. Chapman & Hall/CRC, Boca Raton (2005).
Guo Z.M. (1993) Boundary value problems for a class of quasilinear ordinary differential equations. Diff. Integral Eq. 6:705–719
Kourogenis N.C., Papageorgiou N.S. (2000) Nonsmooth critical point theory and nonlinear elliptic equations at resonance. J. Austral. Math. Soc. Ser. A 69:245–271
Manásevich R.F., Mawhin J. (1998) Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Diff. Eq. 145:367–393
Mawhin J. (2000) Some boundary value problems for Hartman-type perturbations of the ordinary vector p-Laplacian. Nonlinear Anal. 40:497–503
Mawhin, J.: Periodic solutions of systems with p-Laplacian-like operators. In: Nonlinear Analysis and Its Applications to Differential Equations (Lisbon 1998), vol. 43 of Progress in Nonlinear Differential Equations and Their Applications, pp. 37–63, Birkhäuser Verlag, Boston, MA (2001)
Motreanu, D., RPǎdulescu, V.D.: Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems. Kluwer, Dordrecht, (2003).
Papageorgiou E.H., Papageorgiou N.S. (2004) Two nontrivial solutions for quasilinear periodic problems. Proc. Amer. Math. Soc. 132:429–434
Szulkin A. (1986) Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré. Anal. Non Lineaire. 3:77–109
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Gasiński, L. Multiplicity theorems for scalar periodic problems at resonance with p-Laplacian-like operator. J Glob Optim 38, 459–478 (2007). https://doi.org/10.1007/s10898-006-9096-y
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DOI: https://doi.org/10.1007/s10898-006-9096-y