Abstract and Applied Analysis

Existence theorems for elliptic hemivariational inequalities involving the $p$-Laplacian

Nikolaos C. Kourogenis and Nikolaos S. Papageorgiou

Full-text: Open access


We study quasilinear hemivariational inequalities involving the p-Laplacian. We prove two existence theorems. In the first, we allow “crossing” of the principal eigenvalue by the generalized potential, while in the second, we incorporate problems at resonance. Our approach is based on the nonsmooth critical point theory for locally Lipschitz energy functionals.

Article information

Abstr. Appl. Anal., Volume 7, Number 5 (2002), 259-277.

First available in Project Euclid: 14 April 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J85


Kourogenis, Nikolaos C.; Papageorgiou, Nikolaos S. Existence theorems for elliptic hemivariational inequalities involving the $p$-Laplacian. Abstr. Appl. Anal. 7 (2002), no. 5, 259--277. doi:10.1155/S1085337502000908. https://projecteuclid.org/euclid.aaa/1050348437

Export citation


  • W. Allegretto and Y. X. Huang, Eigenvalues of the indefinite-weight $p$-Laplacian in weighted spaces, Funkcial. Ekvac. 38 (1995), no. 2, 233–242.
  • D. Arcoya and L. Orsina, Landesman-Lazer conditions and quasilinear elliptic equations, Nonlinear Anal. 28 (1997), no. 10, 1623–1632.
  • L. Boccardo, P. Drábek, D. Giachetti, and M. Kučera, Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear Anal. 10 (1986), no. 10, 1083–1103.
  • G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), no. 2, 332–336 (Italian).
  • K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), no. 1, 102–129.
  • F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983.
  • D. G. Costa and C. A. Magalhães, Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal. 24 (1995), no. 3, 409–418.
  • A. El Hachimi and J.-P. Gossez, A note on a nonresonance condition for a quasilinear elliptic problem, Nonlinear Anal. 22 (1994), no. 2, 229–236.
  • L. Gasiński and N. S. Papageorgiou, Existence of solutions and of multiple solutions for eigenvalue problems of hemivariational inequalities, Adv. Math. Sci. Appl. 11 (2001), no. 1, 437–464.
  • D. Goeleven, D. Motreanu, and P. D. Panagiotopoulos, Eigenvalue problems for variational-hemivariational inequalities at resonance, Nonlinear Anal. 33 (1998), no. 2, 161–180.
  • S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer Academic Publishers, Dordrecht, 1997.
  • ––––, Handbook of Multivalued Analysis. Volume II: Applications, Kluwer Academic Publishers, Dordrecht, 2000.
  • N. C. Kourogenis and N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, J. Austral. Math. Soc. Ser. A 69 (2000), no. 2, 245–271.
  • P. Lindqvist, On the equation ${\rm div}\,(|\nabla u|\sp {p-2}\nabla u)+\lambda|u| \sp {p-2}u=0$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164.
  • Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, 1995.
  • P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993.
  • P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65, Conference Board of the Mathematical Sciences, American Mathematical Society, Rhode Island, 1986.
  • R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Surveys and Monographs, vol. 49, American Mathematical Society, Rhode Island, 1997.
  • A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 2, 77–109.
  • C.-K. Zhong, On Ekeland's variational principle and a minimax theorem, J. Math. Anal. Appl. 205 (1997), no. 1, 239–250.