Abstract
In this paper, we extend to nonsmooth locally Lipschitz functionals the multiplicity result of Brezis–Nirenberg (Communication Pure Applied Mathematics and 44 (1991)) based on a local linking condition. Our approach is based on the nonsmooth critical point theory for locally Lipschitz functions which uses the Clarke subdifferential. We present two applications. This first concerns periodic systems driven by the ordinary vector p-Laplacian. The second concerns elliptic equations at resonance driven by the partial p-Laplacian with Dirichlet boundary condition. In both cases the potential function is nonsmooth, locally Lipschitz.
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References
P. Bartolo V. Benci D. Fortunato (1983) ArticleTitleAbstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity Nonlinear Analyses 7 981–1012
H. Brezis (1983) Analyse Fonctionnelle Masson Paris
H. Brezis L. Nirenberg (1991) ArticleTitleRemarks on finding critical points Communication Pure Applied Mathematics 44 939–963
G. Cerami (1978) ArticleTitleUn criterio di esistenza per i punti critici su varieta illimitate Rends Academic Science Letters Ist Lobardo 112 332–336
K.-C. Chang (1981) ArticleTitleVariational methods for non-differentiable functionals and their applications to partial differential equations Journal of Math. Anal. Applic. 80 102–129
F.H. Clarke (1983) Optimization and Nonsmooth Analysis Wiley New York
D. Costa J. Goncalves (1990) ArticleTitleCritical point theory for nondifferentiable functionals and applications Journal of Math. Anal. Applic. 153 470–485
M. Degiovanni M. Marzocchi V. Radulescu (2000) ArticleTitleMultiple solutions of hemivariational inequalities with area-type term Calculus of Variations 10 355–387
M. Del Pino R. Manasevich A. Murua (1992) ArticleTitleExistence and multiplicity of solutions with prescribed period for a second order quasilinear ode Nonlinear Analysis 18 79–92 Occurrence Handle10.1016/0362-546X(92)90048-J
P. Drabek S. Invernizzi (1986) ArticleTitleOn the periodic bvp for the forced Duffing equation with jumping nonlinearity Nonlinear Analysis 10 643–650
C. Fabry J. Mawhin M.N. Nkashama (1986) ArticleTitleA multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations Bulletin London Mathematical Society 18 173–180
L. Gasinski N.S. Papageorgiou (2000) ArticleTitleMultiple solutions for nonlinear hemivariational inequalities near resonance Funkcialaj Ekvacioj 43 271–284
L. Gasinski N.S. Papageorgiou (2001) ArticleTitleSolutions and multiple solutions for quasilinear hemivariational inequalities at resonance Proceeding Royal Society Edinburgh (Math) 131A 1–21
D. Goeleven D. Motreanu P. Panagiotopoulos (1998) ArticleTitleEigenvalue problems for variational-hemivariational inequalities at resonance Nonlinear Analysis 33 161–180 Occurrence Handle10.1016/S0362-546X(97)00521-X
S. Hu N.C. Kourogenis N.S. Papageorgiou (1999) ArticleTitleNonlinear elliptic eigenvalue problems with discontinuities Journal of Mathematical Analysis and Applications 233 406–424 Occurrence Handle10.1006/jmaa.1999.6338
S. Hu N.S. Papageorgiou (1997) Handbook of Multivalued Analysis, Volume I: Theory Kluwer Dordrecht, The Netherlands
S. Hu N.S. Papageorgiou (2000) Handbook of Multivalued Analysis, Volume II: Applications Kluwer Dordrecht, The Netherlands
N.C. Kourogenis N.S. Papageorgiou (2000) ArticleTitleNonsmooth critical point theory and nonlinear elliptic equations at resonance Journal of Australian Mathematical Society 69 245–271
Kourogenis, N.C. and Papageorgiou, N.S. (2000). A weak nonsmooth Palais-smale condition and coercivity. Rendiconti del Cir. Matem. Palermo XLIX, 521–526
S. Kyritsi N.S. Papageorgiou (2001) ArticleTitleHemivariational inequalities with the potential crossing the first eigenvalue Bulletin Australian Mathematical Society 64 381–393
S. Kyritsi N.S. Papageorgiou (2005) ArticleTitleNonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities Nonlinear Analyasis 61 373–403
G. Lebourg (1975) ArticleTitleValeur moyenne pour gradient généralisé CRAS, t. 281 792–795
J.Q. Liu S. Li (1984) ArticleTitleExistence theorems of multiple critical points and their applications Kexue Tongbao 17 1025–1027
G. Lieberman (1988) ArticleTitleBoundary regularity for solutions of degenerate elliptic equations Nonlinear Analysis 12 1203–1219 Occurrence Handle10.1016/0362-546X(88)90053-3
P. Lindqvist (1990) ArticleTitleOn the equation div(||Dx||p-2Dx)+λ|x|p-2x=0 Proceeding AMS 109 157–164
J. Mawhin M. Willem (1989) Critical Point Theory and Hamiltonian Systems Springer-Verlag New York
D. Motreanu V. Radulescu (2000) ArticleTitleExistence results for inequality problems with lack of convexity Numerical Functional Analysis Optimization 21 869–884
Z. Naniewicz P. Panagiotopoulos (1995) Mathematical Theory of Hemivariational Inequalities and Applications Marcel Dekker New York
V. Radulescu P. Panagiotopoulos (1998) ArticleTitlePerturbations of hemivariational inequalities with constraints and applications Journal of Global Optimization 12 285–297
M. Struwe (1990) Variational Methods Springer-Verlag Berlin
A. Szulkin (1986) ArticleTitleMinimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems Analyse non Linéaire 3 77–109
C.L. Tang (1998) ArticleTitlePeriodic solutions for nonautonomous second order systems with sublinear nonlinearity Proceedings AMS 126 3263–3270
C.K. Zhong (1997) ArticleTitleOn Ekeland’s variational pronciple and a minimax theorem Journal of Mathematical Analysis and Applications 205 239–250
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Kandilakis, D., Kourogenis, N.C. & Papageorgiou, N.S. “Two Nontrivial Critical Points for Nonsmooth Functionals via Local Linking and Applications”. J Glob Optim 34, 219–244 (2006). https://doi.org/10.1007/s10898-005-3884-7
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DOI: https://doi.org/10.1007/s10898-005-3884-7
Keywords
- Cerami condition
- Critical point
- Generalized subdifferential
- Local linking
- Locally Lipschitz function
- Nonsmooth critical point theory
- Periodic system
- p-Laplacian
- Principal eigenvalue
- Problem at resonance