Abstract
The topic of this chapter is the critical point theory for the functionals that are not locally Lipschitz as was the case in Chatper 2. The setting is more general than in Chatper 2, and the results contain those in Chang [2]. In fact, this chapter presents an extension of Szulkin’s minimax principles [32] for functions of the form I = Φ + Ψ with Φ ∈ C 1(X, ℝ) (X denotes a real Banach space and Ψ a convex, proper and lower semicontinuous functional) to the case where Φ is locally Lipschitz.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Ambrosetti and P.H. Rabinowitz, Dual Variational Methods in Critical Point Theory and Applications, J. Func. Anal. 14 (1973), 349–381.
K.C. Chang, Variational Methods for Nondifferentiable Functionals and Applications to Partial Differential Equations, J. Math. Anal. Appl. 80 (1981), 102–129.
F.H.Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.
Y. Du, A deformation Lemma and some Critical Point Theorems, Bull. Austral. Math. 43 (1991), 161–168.
I. Ekeland, On the Variational Principle, J. Math. Anal. Appl. 47 (1974), 324–353.
I. Ekeland, Nonconvex Minimization Problems, Bull. (New Series) Amer. Math. 1 (1979), 443–474.
N. Ghoussoub, Location, Multiplicity and Morse Indices of Min-Max Critical Points, J. reine angew. Math. 417 (1991), 27–76.
D. Goeleven, V.H. Nguyen and M. Willem, Existence and Multiplicity Results for Semicoercive Unilateral Problems, Bull. Austral. Math. Soc. 49 (1994), 489–497.
D. Goeleven, D. Motreanu and RD.Panagiotopoulos, Multiple Solutions for a Class of Eigenvalue Problems in Hemivariational Inequalities, Nonlinear Anal. TMA 20 (1997), 9–16.
D. Motreanu and P.D.Panagiotopoulos, General Minimax Methods for Variational-Hemivariational Inequalities, submitted.
D. Goeleven and D. Motreanu, Eigenvalue and Dynamic Problems for Variational and Hemivariational Inequalities, Comm. Appl. Nonl. Analysis 3 (1996), 1–21.
L.T. Hu, Homotopy Theory, Academic Press, New York, 1959.
G. Lefter, Critical Point Theorms for Lower Semicontinuous Functions, University of Iasi, unpublished manuscript.
P. Mironescu and V. Radulescu, A Multiplicity Theorem for Locally Lipschitz Periodic functionals, J. Math. Anal. Appl. 195 (1995), 621–637.
D. Motreanu, Existence of Critical Points in a General Setting, Set-Valued Anal. 3 (1995), 295–305.
D. Motreanu, A Multiple Linking Minimax Principle, Bull. Austral. Math. Soc. 53 (1996), 39–49.
D. Motreanu and Z. Naniewicz, Discontinuous Semilinear Problems in Vector-Valued Function Spaces, Differ. Int. Equations 9 (1996), 581–598.
D. Motreanu and P.D.Panagiotopoulos, Nonconvex Energy Functions, Related Eigenvalue Hemivariational Inequalities on the Sphere and Applications, J. Global Optimization 6 (1995), 163–177.
D. Motreanu and P.D. Panagiotopoulos, An Eigenvalue Problem for a Hemivariational Inequality Involving a Nonlinear Compact Operator, Set-Valued Anal. 3 (1995), 157–166.
D. Motreanu and P.D. Panagiotopoulos, A Minimax Approach to the Eigenvalue Problem of Hemivariational Inequalities and Applications, Applicable Anal. 58 (1995), 157–166.
D. Motreanu and P.D.Panagiotopoulos, On the Eigenvalue Problem for Hemivariational Inequalities: Existence and Multiplicity of Solutions, J. Math. Anal. Appl. 197 (1996), 75–89.
D. Motreanu and P.D.Panagiotopoulos, Double Eigenvalue Problems for Hemivariational Inequalities, Arch. Rational Mech. Anal. 140 (1997), 225–251.
J.R. Munkres, Elementary Differential Topology, Princeton University Press, Princeton, 1966.
Z. Naniewicz, Hemivariational Inequalities with Functions Fulfilling Directional Growth Condition, Applicable Anal. 55 (1994), 259–285.
Z. Naniewicz and P.D.Panagiotopoulos, Mathematical Theory of Hemnivariational Inequalities and Applications, Marcel Dekker, Inc., New York, 1995.
P.D.Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Verlag, Basel, Boston, 1985 (Russian Translation MIR Publ. Moscow, 1989).
P.D.Panagiotopoulos, Hemivariational Inequalites. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993.
P.D.Panagiotopoulos, G. Stavroulakis: A Variational — Hemivariational Inequality Approach to the Laminated Plate Theory under Subdifferential Boundary Conditions, Quart. Appl. Math. 46 (1988), 409–430.
PH. Rabinowitz, Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations, in: Nonlinear Analysis: A collection of papers in honor of Erich Röthe, Academic Press, New York, 1978.
P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence, R.I., 1986.
E.H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966.
A. Szulkin, Minimax Principles for Lower Semicontinuous Functions and Applications to Nonlinear Boundary Value Problems, Ann. Inst. Henri Poincaré. Analyse non linéaire 3 (1986), 77–109.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Motreanu, D., Panagiotopoulos, P.D. (1999). Minimax Methods for Variational-Hemivariational Inequalities. In: Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Nonconvex Optimization and Its Applications, vol 29. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4064-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4615-4064-9_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-6820-5
Online ISBN: 978-1-4615-4064-9
eBook Packages: Springer Book Archive