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MULTIPLE SOLUTIONS FOR A DIRICHLET PROBLEM WITH p-LAPLACIAN AND SET-VALUED NONLINEARITY

Part of: Elliptic equations and systems Existence theories

Published online by Cambridge University Press:  01 April 2008

D. AVERNA ,
S. A. MARANO  and
D. MOTREANU
D. AVERNA
Affiliation:
Dipartimento di Matematica ed Applicazioni, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy (email: [email protected])
S. A. MARANO*
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy (email: [email protected])
D. MOTREANU
Affiliation:
Départment de Mathématiques, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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The existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via sub- and supersolution methods as well as variational techniques for nonsmooth functions.

Keywords

p-Laplaciangeneralized gradientmultiple nontrivial solutionssub- and supersolutionscritical points of nonsmooth functions

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations 49J40: Variational methods including variational inequalities
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 2 , April 2008 , pp. 285 - 303
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Ambrosetti, A. and Lupo, D., ‘On a class of nonlinear Dirichlet problems with multiple solutions’, Nonlinear Anal. 8 (1984), 1145–1150.CrossRefGoogle Scholar
[2]Ambrosetti, A. and Mancini, G., ‘Sharp nonuniqueness results for some nonlinear problems’, Nonlinear Anal. 8 (1979), 635–645.CrossRefGoogle Scholar
[3]Carl, S., Le, V. K. and Motreanu, D., Nonsmooth variational problems and their inequalities. comparison principles and applications, Springer Monographs in Mathematics (Springer, New York, 2007).CrossRefGoogle Scholar
[4]Carl, S. and Motreanu, D., ‘Constant-sign and sign-changing solutions for nonlinear eigenvalue problems’, Nonlinear Anal. 68 (2008), 2668–2676.CrossRefGoogle Scholar
[5]Chang, K.-C., ‘Variational methods for non-differentiable functionals and their applications to partial differential equations’, J. Math. Anal. Appl. 80 (1981), 102–129.CrossRefGoogle Scholar
[6]Clarke, F. H., Optimization and nonsmooth analysis, Classics in Applied Mathematics, 5 (SIAM, Philadelphia, PA, 1990).CrossRefGoogle Scholar
[7]Corvellec, J.-N., ‘Morse theory for continuous functionals’, J. Math. Anal. Appl. 196 (1995), 1050–1072.CrossRefGoogle Scholar
[8]Cuesta, M., de Figueiredo, D. and Gossez, J.-P., ‘The beginning of the Fučik spectrum for the p-Laplacian’, J. Differential Equations 159 (1999), 212–238.CrossRefGoogle Scholar
[9]Gasiński, L. and Papageorgiou, N. S., Nonsmooth critical point theory and nonlinear boundary value problems (Chapman & Hall/CRC, Boca Raton, FL, 2005).Google Scholar
[10]Gasiński, L. and Papageorgiou, N. S., Topics in nonlinear analysis (Chapman & Hall/CRC, Boca Raton, FL, 2006).Google Scholar
[11]Marano, S. A., Molica Bisci, G. and Motreanu, D., ‘Multiple solutions for a class of elliptic hemivariational inequalities’, J. Math. Anal. Appl. 337 (2008), 85–97.CrossRefGoogle Scholar
[12]Marano, S. A. and Papageorgiou, N. S., ‘On some elliptic hemivariational and variational-hemivariational inequalities’, Nonlinear Anal. 62 (2005), 757–774.CrossRefGoogle Scholar
[13]Motreanu, D., Motreanu, V. V. and Pasca, D., ‘A version of Zhong’s coercivity result for a general class of nonsmooth functionals’, Abstr. Appl. Anal. 7 (2002), 601–612.CrossRefGoogle Scholar
[14]Motreanu, D. and Panagiotopoulos, P. D., Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, Nonconvex Optimization and its Applications, 29 (Kluwer, Dordrecht, 1998).Google Scholar
[15]Motreanu, D. and Papageorgiou, N. S., ‘Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential’, Nonlinear Anal. 56 (2004), 1211–1234.CrossRefGoogle Scholar
[16]Motreanu, D. and Radulescu, V., Variational and non-variational methods in nonlinear analysis and boundary value problems, Nonconvex Optimization and its Applications, 67 (Kluwer, Dordrecht, 2003).CrossRefGoogle Scholar
[17]Papageorgiou, E. H. and Papageorgiou, N. S., ‘A multiplicity theorem for problems with the p-Laplacian’, J. Funct. Anal. 244 (2007), 63–77.CrossRefGoogle Scholar
[18]Struwe, M., ‘A note on a result of Ambrosetti and Mancini’, Ann. Mat. Pura Appl. 131 (1982), 107–115.CrossRefGoogle Scholar
[19]Struwe, M., Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, 2nd edn (Springer, Berlin, 1996).Google Scholar