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Xiao, Yi-Bin; Huang, Nan-Jing

Sub-supersolution method and extremal solutions for higher order quasi-linear elliptic hemi-variational inequalities. (English) Zbl 1110.49012

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 8, 1739-1752 (2007).
Summary: We generalize the sub-supersolution method for a class of higher order quasi-linear elliptic hemi-variational inequalities. Using the notion of sub and supersolution, we prove the existence, comparison, compactness and extremality results for the higher order quasi-linear elliptic hemi-variational inequality under considerations.

Cited in 8 Documents

MSC:

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
35J50 Variational methods for elliptic systems

Keywords:

subsolution; supersolution; extremal solution; hemi-variational inequality; \(p\)-Laplacian operator
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\textit{Y.-B. Xiao} and \textit{N.-J. Huang}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 8, 1739--1752 (2007; Zbl 1110.49012)
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References:

[1] Akô, K., On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. math. soc. Japan, 13, 45-62, (1961) · Zbl 0102.09303
[2] Carl, S., Existence and comparison results for variatinoal – hemivariational inequalities, J. inequal. appl., 2005:1, 33-40, (2005) · Zbl 1080.49007
[3] Carl, S.; Heikkilä, S., Nonlinear differential equations in ordered spaces, (2000), Chapman & Hall/CRC Boca Raton, FL · Zbl 0948.34001
[4] Carl, S.; Le, V.K.; Motreanu, D., The sub – supersolution method and extremal solutions for quasilinear hemivariational inequalities, Differential integral equations, 17, 165-178, (2004) · Zbl 1164.35301
[5] Carl, S.; Le, V.K.; Motreanu, D., Existence and comparison results for quasilinear evolution hemivariational inequalities, Electron. J. differential equations, 57, 1-17, (2004) · Zbl 1053.49005
[6] Chang, K.C., Critical point theorem and its applications, (1986), Shanghai Press of Science and Technology Shanghai, (in Chinese)
[7] Clarke, F.H., Optimization and nonsmooth analysis, (1990), SIAM Philadelphia · Zbl 0727.90045
[8] Deuel, J.; Hess, P., A criterion for the existence of solutions of non-linear elliptic boundary value problems, Proc. roy. soc. Edinburgh sect. A, 74, 49-54, (1974/1975) · Zbl 0331.35028
[9] Deuel, J.; Hess, P., Nonlinear parabolic boundary value problems with upper and lower solutions, Israel J. math., 29, 92-104, (1978) · Zbl 0372.35045
[10] Evans, L.C., Partial differential equations, (1998), Amer. Math. Soc. Providence, Rhode Island
[11] Le, V.K., Subsolution – supersolutions method in variational inequalities, Nonlinear anal., 45, 775-800, (2001) · Zbl 1040.49008
[12] Le, V.K., Subsolution – supersolutions and the existence of extremal solutions in noncoercive variational inequalities, JIPAM J. inequal. pure appl. math., 2, 2, (2001), (article 20 (electronic))
[13] Liu, Z.H., On quasilinear elliptic hemivariational inequalities, Appl. math. mech., 20, 2, 225-230, (1999) · Zbl 0932.49010
[14] Liu, Z.H., A class of quasilinear elliptic hemivariational inequalities, Acta math. appl. sin., 17, 2, 279-285, (2001) · Zbl 1005.35049
[15] Liu, Z.H., Generalized quasi-variational hemi-variational inequalities, Appl. math. lett., 17, 741-745, (2004) · Zbl 1058.49006
[16] Liu, Z.H.; Simon, L., Existence results for evolution hemivariational inequalities, Adv. math., 30, 1, 47-55, (2001) · Zbl 0983.49004
[17] Mustonen, V., Mappings of monotone type: theory and applications, (), 104-126
[18] Motreanu, D.; Panagiotopoulos, P.D., Minimax theorems and qualitative properties of the solutions of hemivariational inequalities and applications, () · Zbl 0829.49005
[19] Naniewicz, Z.; Panagiotopoulos, P.D., Mathematical theory of hemivariational inequalities and applications, (1995), Marcel Dekker New York · Zbl 0968.49008
[20] Panagiotopoulos, P.D., Hemivariational inequalities. applications in mechanics and engineering, (1993), Springer-Verlag Berlin · Zbl 0826.73002
[21] Panagiotopoulos, P.D., Coercive and semicoercive hemivariational inequalities, Nonlinear anal., 16, 209-231, (1991) · Zbl 0733.49012
[22] Panagiotopoulos, P.D., Hemivariational inequalitiy and Fan-variational inequality. new applications and results, Atti. sem. mat. fis. univ. modena, XLIII, 159-191, (1995) · Zbl 0843.49006
[23] Panagiotopoulos, P.D.; Fundo, M.; Radulescu, V., Existence theorems of hartman – stampacchia type for hemivariational inequalities and applications, J. global optim., 15, 41-54, (1999) · Zbl 0951.49018
[24] Sattinger, D.H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana univ. math. J., 21, 979-1000, (1972) · Zbl 0223.35038
[25] Showalter, R.E., Monotone operators in Banach space and nonlinear partial diffrential equations, (1997), Amer. Math. Soc. Providence · Zbl 0870.35004
[26] Verma, R.U., Nonlinear variational and constrained hemivariational inequalities involving relaxed operators, Z. angew. math. mech., 77, 5, 387-391, (1997) · Zbl 0886.49006
[27] Zeidler, E., ()
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