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Solutions and multiple solutions for superlinear perturbations of the periodic scalar p-Laplacian

Part of: Boundary value problems Qualitative theory Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  28 June 2013

Sophia Th. Kyritsi ,
Donal O'Regan  and
Nikolaos S. Papageorgiou
Sophia Th. Kyritsi
Affiliation:
Department of Mathematics, Hellenic Naval Academy, Piraeus 18539, Greece (skyrits@math.ntua.gr)
Donal O'Regan
Affiliation:
Department of Mathematics, National University of Ireland, Galway, Ireland (donal.oregan@nuigalway.ie)
Nikolaos S. Papageorgiou
Affiliation:
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece (npapg@math.ntua.gr)
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Abstract

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We consider a nonlinear periodic problem driven by the scalar p-Laplacian and with a reaction term which exhibits a (p – 1)-superlinear growth near ±∞ but need not satisfy the Ambrosetti-Rabinowitz condition. Combining critical point theory with Morse theory we prove an existence theorem. Then, using variational methods together with truncation techniques, we prove a multiplicity theorem establishing the existence of at least five non-trivial solutions, with precise sign information for all of them (two positive solutions, two negative solutions and a nodal (sign changing) solution).

Keywords

scalar p-Laplaciancritical groupsmountain pass theoremC conditionp-superlinearityAR condition

MSC classification

Secondary: 34B15: Nonlinear boundary value problems 34B18: Positive solutions of nonlinear boundary value problems 34C25: Periodic solutions 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 805 - 827
Copyright
Copyright © Edinburgh Mathematical Society 2013 

References

1.Aizicovici, S., Papageorgiou, N. S. and Staicu, V., Periodic solutions for second order differential inclusions with the scalar p-Laplacian, J. Math. Analysis Applic. 322 (2006), 913–929.CrossRefGoogle Scholar
2.Aizicovici, S., Papageorgiou, N. S. and Staicu, V., Multiple nontrivial solutions for nonlinear periodic problems with the p-Laplacian, J. Diff. Eqns 243 (2007), 504–535.CrossRefGoogle Scholar
3.Aizicovici, S., Papageorgiou, N. S. and Staicu, V., Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Memoirs of the American Mathematical Society, Volume 196 (American Mathematical Society, Providence, RI, 2008).Google Scholar
4.Aizicovici, S., Papageorgiou, N. S. and Staicu, V., Existence of multiple solutions for superlinear Neumann problems, Annali Mat. Pura Appl. 188 (2009), 679–719.CrossRefGoogle Scholar
5.Ambrosetti, A. and Rabinowitz, P., Dual variational methods in the critical point theory and applications, J. Funct. Analysis 14 (1973), 349–381.CrossRefGoogle Scholar
6.Bartolo, P., Benci, V. and Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with ‘strong’ resonance at infinity, Nonlin. Analysis 7 (1983), 981–1012.CrossRefGoogle Scholar
7.Binding, P. A. and Rynne, B. P., The spectrum of the periodic p-Laplacian, J. Diff. Eqns 235 (2007), 199–218.CrossRefGoogle Scholar
8.Chang, K. C., Infinite dimensional Morse theory and multiple solution problems (Birkhäuser, Boston, MA, 1993).CrossRefGoogle Scholar
9.Chang, K. C., Methods in nonlinear analysis (Springer, 2005).Google Scholar
10.Cingolani, S. and Degiovanni, M., Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Commun. PDEs 30 (2005), 1191–1203.CrossRefGoogle Scholar
11.del Pino, M., Manásevich, M. A. R. and Murúa, A., Existence and multiplicity of solutions with prescribed period for second order quasilinear ODEs, Nonlin. Analysis 18 (1992), 79–92.CrossRefGoogle Scholar
12.Drabek, P. and Manásevich, R., On the closed solution to some nonhomogeneous eigen-value problems with p-Laplacian, Diff. Integ. Eqns 12 (1999), 773–788.Google Scholar
13.Dugundji, J., Topology (Allyn and Bacon, Boston, MA, 1966).Google Scholar
14.Fadell, E. R. and Rabinowitz, P., Generalized cohomological index theories for Lie groups actions with applications to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139–174.CrossRefGoogle Scholar
15.Gasiński, L. and Papageorgiou, N. S., Three nontrivial solutions for periodic problems with the p-Laplacian and a p-superlinear nonlinearity, Commun. Pure Appl. Analysis 8 (2009), 1421–1437.CrossRefGoogle Scholar
16.Granas, A. and Dugundji, J., Fixed point theory (Springer, 2003).CrossRefGoogle Scholar
17.Jiang, M. Y. and Wang, Y., Solvability of the resonant 1-dimensional periodic p-Laplacian, J. Math. Analysis Applic. 370 (2010), 107–131.CrossRefGoogle Scholar
18.Li, G. and Yang, C., The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti–Rabinowitz condition, Nonlin. Analysis 72 (2010), 4602–4613.CrossRefGoogle Scholar
19.Miyagaki, O. and Souto, M. A. S., Superlinear problems without Ambrosetti–Rabinowitz growth condition, J. Diff. Eqns 245 (2008), 3628–3638.CrossRefGoogle Scholar
20.Motreanu, D., Motreanu, V. and Papageorgiou, N. S., Multiple solutions for resonant periodic equations, Nonlin. Diff. Eqns Applic. 17 (2010), 535–557.CrossRefGoogle Scholar
21.Papageorgiou, N. S. and Kyritsi-Yiallourou, S. Th., Handbook of applied analysis (Springer, 2009).Google Scholar
22.Papageorgiou, E. and Papageorgiou, N. S., Two nontrivial solutions for quasilinear periodic problems, Proc. Am. Math. Soc. 132 (2004), 429–434.CrossRefGoogle Scholar
23.Perera, K., Agarwal, R. and O’Regan, D., Morse theoretic aspects of p-Laplacian type operators, Mathematical Surveys and Monographs, Volume 161 (American Mathematical Society, Providence, RI, 2010).CrossRefGoogle Scholar
24.Rynne, B. P., p-Laplacian problems with jumping nonlinearities, J. Diff. Eqns 226 (2006), 501–524.CrossRefGoogle Scholar
25.Vázquez, J. L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.CrossRefGoogle Scholar