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Keywords:
subdifferential; critical point; Palais-Smale condition; Mountain Pass Theorem; Saddle Point Theorem; multivalued term; Dirichlet problem; Neumann problem; p-Laplacian; Rayleigh quotient
subdifferential; critical point; Palais-Smale condition; Mountain Pass Theorem; Saddle Point Theorem; multivalued term; Dirichlet problem; Neumann problem; p-Laplacian; Rayleigh quotient
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.
References:
[1] R. Adams: Sobolev Spaces. Academic Press, New York, 1975. MR 0450957 | Zbl 0314.46030
[2] W. F. Ames: Nonlinear Partial Differential Equations in Engineering. Academic Press, New York, 1965. MR 0210342 | Zbl 0176.39701
[3] A. Ambrosetti and P. H. Rabinowitz: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381. DOI 10.1016/0022-1236(73)90051-7 | MR 0370183
[4] A. Anane and J. P. Gossez: Strongly nonlinear elliptic problems near resonance: a variational approach. Comm. Partial Differential Equations 15 (1990), 1141–1159. DOI 10.1080/03605309908820717 | MR 1070239
[5] D. Arcoya and M. Calahorrano: Some discontinuous problems with a quasilinear operator. J. Math. Anal. Appl. 187 (1994), 1059–1072. DOI 10.1006/jmaa.1994.1406 | MR 1298837
[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
[7] K. C. Chang: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80 (1981), 102–129. DOI 10.1016/0022-247X(81)90095-0 | MR 0614246
[8] D. Costa and C. Magalhaes: Existence results for perturbations of the p-Laplacian. Nonlinear Anal. TMA 24 (1995), 409–418. MR 1312776
[9] C. De Coster: Pairs of positive solutions for the one-dimensional p-Laplacian. Nonlinear Anal. TMA 23 (1994), 669–681. MR 1297285 | Zbl 0813.34021
[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. DOI 10.1016/0022-0396(89)90093-4 | MR 1003248
[11] A. Friedman: Generalized heat transfer between solids and gases under nonlinear boundary conditions. J. Math. Mech. 8 (1959), 161–184. MR 0102345 | Zbl 0101.31102
[12] Z. Guo: Boundary value problems for a class of quasilinear ordinary differential equations. Differential Integral Equations 6 (1993), 705–719. MR 1202567
[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
[14] S. Hu and N. S. Papageorgiou: Handbook of Multivalued Analysis Volume I: Theory. Kluwer Academic Publishers, Dordrecht, 1997. MR 1485775
[15] A. Ioffe and V. Tichomirov: Theory of Extremal Problems. North Holland, Amsterdam, 1979. MR 0528295
[16] N. Kenmochi: Pseudomonotone operators and nonlinear elliptic boundary value problems. J. Math. Soc. Japan 27 (1975), 121–149. DOI 10.2969/jmsj/02710121 | MR 0372419 | Zbl 0292.35034
[17] A. Kufner, O. John and S. Fučík: Function Spaces. Noordhoff, Leyden, The Netherlands, 1977. MR 0482102
[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
[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. MR 0501092 | Zbl 0466.58015
[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. MR 0845785 | Zbl 0609.58002
[21] R. Showalter: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Math. Surveys, vol. 49, AMS, Providence, R. I., 1997. MR 1422252 | Zbl 0870.35004
[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. DOI 10.1016/S0294-1449(16)30389-4 | MR 0837231 | Zbl 0612.58011
[23] E. Zeidler: Nonlinear Functional Analysis and its Applications II. Springer Verlag, New York, 1990. MR 0816732 | Zbl 0684.47029
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