Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T11:40:12.638Z Has data issue: false hasContentIssue false

Hemivariational inequalities with the potential crossing the first eigenvalue

Published online by Cambridge University Press:  17 April 2009

Sophia Th. Kyritsi
Affiliation:
National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
Nikolaos S. Papageorgiou
Affiliation:
National Technical University, Department of Mathematics, Zografou Campus, Athens 157 80, Greece e-mail:[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study a nonlinear hemivariational inequality involving the p-Laplacian. Our approach is variational and uses a recent nonsmooth Linking Theorem, due to Kourogenis and Papageorgiou (2000). The use of the Linking Theorem instead of the Mountain Pass Theorem allows us to assume an asymptotic behaviour of the generalised potential function which goes beyond the principal eigenvalue of the negative p-Laplacian with Dirichlet boundary conditions.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Anane, A., ‘Simplicite et isolation de la premier valeur proper du p-Laplacien avec poids’, C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), 725728.Google Scholar
[2]Anane, A. and Tsouli, N., ‘On the second eigenvalue of the p-Laplacian’, in Nonlinear Partial Differential equations, Pitman Research Notes Math. Ser. 343 (Longman, Harlow, 1996), pp. 19.Google Scholar
[3]Bartolo, P., Benci, V. and Fortunato, D., ‘Abstract critical point theorems and applications to some nonlinear problems with strong resonance’, Nonlinear Anal. 7 (1983), 9811012.CrossRefGoogle Scholar
[4]Cerami, G., ‘An existence criterion for the critical points on unbounded manifolds’, (in Italian), Instit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), 332336.Google Scholar
[5]Chang, K.C., ‘Variational methods for nondifferentiable functionals and their applications to partial differential equations’, J. Math. Anal. Appl. 80 (1981), 102129.CrossRefGoogle Scholar
[6]Clarke, F.H., Optimization and nonsmooth analysis (Wiley, New York, 1983).Google Scholar
[7]Costa, D.G. and Magalhaes, C. A., ‘Existence results for perturbations of the p-Laplacian’, Nonlinear Anal. 24 (1995), 409418.CrossRefGoogle Scholar
[8]Diaz, J.I., Nonlinear partial differential equations and free boundaries Volume 1: Elliptic equations, Research Notes in Math. 106 (Pitman, London, 1985).Google Scholar
[9]Gasinski, L. and Papageorgiou, N.S., ‘Nonlinear hemivariational inequalities at resonance’, Bull. Austral. Math. Soc. 60 (1999), 353364.CrossRefGoogle Scholar
[10]Gasinski, L. and Papageorgiou, N.S., ‘An existence theorem for nonlinear hemivariational inequalities at resonance’, Bull. Austral. Math. Soc. 63 (2001), 114.CrossRefGoogle Scholar
[11]Gasinski, L. and Papageorgiou, N.S., ‘Multiple solutions for nonlinear hemivariational inequalities near resonance’, Funkcial. Ekvac. 43 (2000), 271284.Google Scholar
[12]Gasinski, L. and Papageorgiou, N.S., ‘Existence of solutions and of multiple solutions for eigenvalue problems of hemivariational inequalities’, Adv. Math. Sci. App (to appear).Google Scholar
[13]Gasinski, L. and Papageorgiou, N.S., ‘Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance’, Proc. Royal Soc. Edinburgh 131A (2001), 121.Google Scholar
[14]Goeleven, D., Motreanu, D. and Panagiotopoulos, P., ‘Multiple solutions for a class of eigenvalue problems in hemivariational inequalities’, Nonlinear Anal. 29 (1997), 926.CrossRefGoogle Scholar
[15]Goeleven, D., Motreanu, D. and Panagiotopoulos, P., ‘Eigenvalue problems for variational-hemivariational inequalities at resonance’, Nonlinear Anal. 33 (1998), 161180.CrossRefGoogle Scholar
[16]Hu, S. and Papageorgion, N.S., Handbook of multivalued analysis Volume I: Theory, Mathematics and its Applications 419 (Kluwer, Drodrecht, 1997).CrossRefGoogle Scholar
[17]Hu, S. and Papageorgiou, N.S., Handbook of multivalued analysis Volume II: Applications, Mathematics and its Applications 500 (Kluwer, Drodrecht, 2000).CrossRefGoogle Scholar
[18]Kourogenis, N. and Papageorgiou, N.S., ‘Nonsmooth critical point theory and nonlinear elliptic equations at resonance’, J. Austral. Math. Soc. Ser. A 69 (2000), 245271.CrossRefGoogle Scholar
[19]Lebourg, G., ‘Valeur moyenne pour gradient généralisé’, C.R. Acad. Sci. Paris Ser. A-B 281 (1975), 795797.Google Scholar
[20]Lieberman, G.M., ‘Boundary regularity for solutions of degenerate elliptic equations’, Nonlinear Anal. 12 (1998), 12031219.CrossRefGoogle Scholar
[21]Lindqvist, P., ‘On the equation div(|Dx|p–2Dx) + λ|x|p–2x = 0’, Proc. Amer. Math. Soc. 109 (1991), 157164.Google Scholar
[22]Naniewicz, Z. and Panagiotopoulos, P., Mathematical theory of hemivariational inequalities and applications, Monographs and Textbooks in Pure and Applied Mathematics 188 (Marcel-Dekker, New York, 1995).Google Scholar
[23]Panagiotopoulos, P., Hemivariational inequalities. Applications to mechanics and engineering (Springer-Verlag, New York, 1993).CrossRefGoogle Scholar
[24]Struwe, M., Variational methods. Applications to nolinear partical differential equations and Hamiltonian systems, A series of modern surveys in Mathematics 34 (Springer-Verlag, Berlin, 1990).Google Scholar