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## Colloquium Mathematicum

1999 | 81 | 1 | 89-99
Tytuł artykułu

### Multiple solutions for nonlinear discontinuous elliptic problems near resonance

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when $λ → λ_1$ from the left, $λ_1$ being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
89-99
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-09-16
poprawiono
1999-01-28
Twórcy
• Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece
• Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece
Bibliografia
• [1] Ambrosetti, A., Garcia Azorero, J. and Peral, I.: Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219-242.
• [2] Ambrosetti, A. and Rabinowitz, P.: Dual variational methods in critical point theory and applications, ibid. 14 (1973), 349-381.
• [3] Chang, K. C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129.
• [4] Chiappinelli, R. and De Figueiredo, D.: Bifurcation from infinity and multiple solutions for an elliptic system, Differential Integral Equations 6 (1993), 757-771.
• [5] Chiappinelli, R., Mawhin, J. and Nugari, R.: Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities, Nonlinear Anal. 18 (1992), 1099-1112.
• [6] Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
• [7] De Figueiredo, D.: The Ekeland Variational Principle with Applications and Detours, Springer, Berlin, 1989.
• [8] Hu, S. and Papageorgiou, N. S.: Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, 1997.
• [9] Kenmochi, N.: Pseudomonotone operators and nonlinear elliptic boundary value problems, J. Math. Soc. Japan 27 (1975), 121-149.
• [10] Kourogenis, N. C. and Papageorgiou, N. S.: Discontinuous quasilinear elliptic problems at resonance, Colloq. Math. 78 (1998), 213-223.
• [11] Lindqvist, P.: On the equation $div(∥Dx∥^p-2Dx) +λ|x|^p-2x=0$, Proc. Amer. Math. Soc. 109 (1991), 157-164.
• [12] Mawhin, J. and Schmitt, K.: Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results Math. 14 (1988), 138-146.
• [13] Mawhin, J. and Schmitt, K.: Nonlinear eigenvalue problems with a parameter near resonance, Ann. Polon. Math. 51 (1990), 241-248.
• [14] Rabinowitz, R.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. 65, Amer. Math. Soc., Providence, RI, 1986.
• [15] Ramos, M. and Sanchez, L.: A variational approach to multiplicity in elliptic problems near resonance, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 385-394.
• [16] Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150.
Typ dokumentu
Bibliografia
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