Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-21T05:27:00.159Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A DIRICHLET PROBLEM WITH p-LAPLACIAN AND SET-VALUED NONLINEARITY

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  14 October 2011

S. A. MARANO
S. A. MARANO*
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy (email: marano@dmi.unict.it)
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.

The existence of solutions to a homogeneous Dirichlet problem for a p-Laplacian differential inclusion is studied via a fixed-point type theorem concerning operator inclusions in Banach spaces. Some meaningful special cases are then worked out.

Keywords

p-Laplaciandifferential inclusiongeneralized gradientoperator inclusion

MSC classification

Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 83 - 89
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Averna, D. and Marano, S. A., ‘Existence theorems for inclusions of the type Ψ(u)(t)∈F(t,Φ(u)(t))’, Appl. Anal. 72 (1999), 449–458.CrossRefGoogle Scholar
[2]Averna, D., Marano, S. A. and Motreanu, D., ‘Multiple solutions for a Dirichlet problem with p-Laplacian and set-valued nonlinearity’, Bull. Aust. Math. Soc. 77 (2008), 285–303.CrossRefGoogle Scholar
[3]Bonanno, G. and Molica Bisci, G., ‘Infinitely many solutions for a Dirichlet problem involving the p-Laplacian’, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 737–752.CrossRefGoogle Scholar
[4]Bouguima, S. M., ‘A quasilinear elliptic problem with a discontinuous nonlinearity’, Nonlinear Anal. 25 (1995), 1115–1122.CrossRefGoogle Scholar
[5]Brézis, H., Analyse Fonctionnelle—Théorie et Applications (Masson, Paris, 1983).Google Scholar
[6]Burenkov, V. I. and Gusakov, V. A., ‘On precise constants in Sobolev imbedding theorems’, Sov. Math. Dokl. 35 (1987), 651–655.Google Scholar
[7]Carl, S. and Heikkilä, S., ‘p-Laplacian inclusions via fixed points for multifunctions in posets’, Set-Valued Anal. 16 (2008), 637–649.CrossRefGoogle Scholar
[8]Carl, S., Le, V. K. and Motreanu, D., Nonsmooth Variational Problems and Their Inequalities (Springer, New York, 2007).CrossRefGoogle Scholar
[9]Chabrowski, J., Variational Methods for Potential Operator Equations, de Gruyter Series in Nonlinear Analysis and Applications, 24 (de Gruyter, Berlin, 1997).CrossRefGoogle Scholar
[10]Gasiński, L. and Papageorgiou, N. S., Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Series in Mathematical Analysis and Applications, 8 (Chapman and Hall/CRC Press, Boca Raton, 2005).Google Scholar
[11]Lindqvist, P., ‘On the equation div(∣∇u∣p−2∇u)+λ∣u∣p−2u=0’, Proc. Amer. Math. Soc. 109 (1990), 157–164, Addendum: Proc. Amer. Math. Soc. 116 (1992), 583–584.Google Scholar
[12]Livrea, R. and Marano, S. A., ‘Non-smooth critical point theory’, in: Handbook of Nonconvex Analysis and Applications (eds. Gao, D. Y. and Motreanu, D.) (International Press of Boston, Somerville, 2010), pp. 353–407.Google Scholar
[13]Marano, S. A., ‘Existence theorems for a semilinear elliptic boundary value problem’, Ann. Polon. Math. 60 (1994), 57–67.CrossRefGoogle Scholar
[14]Marano, S. A. and Papageorgiou, N. S., ‘On some elliptic hemivariational and variational-hemivariational inequalities’, Nonlinear Anal. 62 (2005), 757–774.CrossRefGoogle Scholar
[15]Peral, I., ‘Multiplicity of solutions for the p-Laplacian’, in: ICTP Lecture Notes of the Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997.Google Scholar
[16]Talenti, G., ‘Some inequalities of Sobolev type on two-dimensional spheres’, in: General Inequalities 5, International Series of Numerical Mathematics, 80 (ed. Walter, W.) (Birkhäuser, Basel, 1987), pp. 401–408.CrossRefGoogle Scholar