Previous |  Up |  Next

Article

Title: Quasilinear elliptic problems with multivalued terms (English)
Author: Halidias, Nikolaos
Author: Papageorgiou, Nikolaos S.
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 50
Issue: 4
Year: 2000
Pages: 803-823
Summary lang: English
.
Category: math
.
Summary: We study the quasilinear elliptic problem with multivalued terms.We consider the Dirichlet problem with a multivalued term appearing in the equation and a problem of Neumann type with a multivalued term appearing in the boundary condition. Our approach is based on Szulkin’s critical point theory for lower semicontinuous energy functionals. (English)
Keyword: subdifferential
Keyword: critical point
Keyword: Palais-Smale condition
Keyword: Mountain Pass Theorem
Keyword: Saddle Point Theorem
Keyword: multivalued term
Keyword: Dirichlet problem
Keyword: Neumann problem
Keyword: p-Laplacian
Keyword: Rayleigh quotient
MSC: 35J20
MSC: 35J60
idZBL: Zbl 1079.35511
idMR: MR1792971
.
Date available: 2009-09-24T10:38:00Z
Last updated: 2020-07-03
Stable URL: http://hdl.handle.net/10338.dmlcz/127611
.
Reference: [1] R. Adams: Sobolev Spaces.Academic Press, New York, 1975. Zbl 0314.46030, MR 0450957
Reference: [2] W. F. Ames: Nonlinear Partial Differential Equations in Engineering.Academic Press, New York, 1965. Zbl 0176.39701, MR 0210342
Reference: [3] A. Ambrosetti and P. H. Rabinowitz: Dual variational methods in critical point theory and applications.J. Funct. Anal. 14 (1973), 349–381. MR 0370183, 10.1016/0022-1236(73)90051-7
Reference: [4] A. Anane and J. P. Gossez: Strongly nonlinear elliptic problems near resonance: a variational approach.Comm. Partial Differential Equations 15 (1990), 1141–1159. MR 1070239, 10.1080/03605309908820717
Reference: [5] D. Arcoya and M. Calahorrano: Some discontinuous problems with a quasilinear operator.J. Math. Anal. Appl. 187 (1994), 1059–1072. MR 1298837, 10.1006/jmaa.1994.1406
Reference: [6] L. Boccardo, P. Drábek, D. Giachetti and M. Kučera: Generalization of Fredholm alternative for nonlinear differential operators.Nonlinear Anal. TMA 10 (1986), 1083–1103. MR 0857742
Reference: [7] K. C. Chang: Variational methods for nondifferentiable functionals and their applications to partial differential equations.J. Math. Anal. Appl. 80 (1981), 102–129. MR 0614246, 10.1016/0022-247X(81)90095-0
Reference: [8] D. Costa and C. Magalhaes: Existence results for perturbations of the p-Laplacian.Nonlinear Anal. TMA 24 (1995), 409–418. MR 1312776
Reference: [9] C. De Coster: Pairs of positive solutions for the one-dimensional p-Laplacian.Nonlinear Anal. TMA 23 (1994), 669–681. Zbl 0813.34021, MR 1297285
Reference: [10] M. Del Pino, M. Elgueta and R. Manasevich: A homotopic deformation along p of a Leray-Shauder degree result and existence for $(|u^{\prime }|^{p-2}u^{\prime })^{\prime }+f(t,u) = 0$, $ u(0)=u(T)=0$, $p>1$.J. Differential Equations 80 (1989), 1–13. MR 1003248, 10.1016/0022-0396(89)90093-4
Reference: [11] A. Friedman: Generalized heat transfer between solids and gases under nonlinear boundary conditions.J. Math. Mech. 8 (1959), 161–184. Zbl 0101.31102, MR 0102345
Reference: [12] Z. Guo: Boundary value problems for a class of quasilinear ordinary differential equations.Differential Integral Equations 6 (1993), 705–719. MR 1202567
Reference: [13] A. El. Hachimi, J.-P. Gossez: A note on a nonresonance condition for a quasilinear elliptic problem.Nonlinear Anal. TMA 22 (1994), 229–236. MR 1258959
Reference: [14] S. Hu and N. S. Papageorgiou: Handbook of Multivalued Analysis Volume I: Theory.Kluwer Academic Publishers, Dordrecht, 1997. MR 1485775
Reference: [15] A. Ioffe and V. Tichomirov: Theory of Extremal Problems.North Holland, Amsterdam, 1979. MR 0528295
Reference: [16] N. Kenmochi: Pseudomonotone operators and nonlinear elliptic boundary value problems.J. Math. Soc. Japan 27 (1975), 121–149. Zbl 0292.35034, MR 0372419, 10.2969/jmsj/02710121
Reference: [17] A. Kufner, O. John and S. Fučík: Function Spaces.Noordhoff, Leyden, The Netherlands, 1977. MR 0482102
Reference: [18] P. Lindqvist: On the equation $\div (|Dx|^{p-2}Dx)+ \lambda |x|^{p-2}x = 0$.Proc. AMS vol. 109, 1991, pp. 157–164. MR 1007505
Reference: [19] P. H. Rabinowitz: Some minimax theorems and applications to nonlinear partial differential equations.Nonlinear Analysis: A collection of papers of E. Rothe, L. Cesari, R. Kannan, H. F. Weinberger (eds.), Acad. Press, New York, 1978, pp. 161–177. Zbl 0466.58015, MR 0501092
Reference: [20] P. H. Rabinowitz: Minimax Methods in Critical Point Theory with Applications to Differential Equations.CBMS, Regional Conference Series in Math, No 65, AMS, Providence, R. J., 1986. Zbl 0609.58002, MR 0845785
Reference: [21] R. Showalter: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations.Math. Surveys, vol. 49, AMS, Providence, R. I., 1997. Zbl 0870.35004, MR 1422252
Reference: [22] A. Szulkin: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems.Ann. Inst. H. Poincare Anal. Non Linéaire 3 (1986), 77–109. Zbl 0612.58011, MR 0837231, 10.1016/S0294-1449(16)30389-4
Reference: [23] E. Zeidler: Nonlinear Functional Analysis and its Applications II.Springer Verlag, New York, 1990. Zbl 0684.47029, MR 0816732
.

Files

Files Size Format View
CzechMathJ_50-2000-4_9.pdf 432.7Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo