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On some noncoercive variational inequalities containing degenerate elliptic operators

Published online by Cambridge University Press:  17 February 2009

Vy Khoi Le
Vy Khoi Le
Affiliation:
Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO 65409, USA; e-mail: vy@umr.edu.
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Abstract

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We are concerned with the solvability of variational inequalities that contain degenerate elliptic operators. By using a recession approach, we find conditions on the boundary conditions such that the inequality has at least one solution. Existence results of Landesman-Lazer type for a nonsmooth inequality and a resonance problem for a weighted p-Laplacian are discussed in detail.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 3 , January 2003 , pp. 409 - 430
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Adams, R., Sobolev spaces (Academic Press, New York, 1975).Google Scholar
[2]Alvino, A. and Trombetti, G., “Sulle migliori di maggiorazione per une classe di equazioni ellittiche degeneri”, Ric. Mat. 27 (1978) 413–428.Google Scholar
[3]Alvino, A. and Trombetti, G., “Su une classe di equazioni ellittiche non lineari degeneri”, Ric. Mat. 29 (1980) 193–212.Google Scholar
[4]Anane, A., “Simplicité et isolation de la première valeur propre du p-Laplacien”, C. R. Acad. Sci Paris 305 (1987) 725–728.Google Scholar
[5]Ang, D. D., Schmitt, K. and Le, V. K., “Noncoercive variational inequalities: Some applications”, Nonlinear Analysis, TMA 15 (1990) 497–512.Google Scholar
[6]Ang, D. D., Schmitt, K. and Le, V. K., “P-coercive variational inequalities and unilateral problems for von Karman's equations”, WSSIAA 1 (1992) 15–29.Google Scholar
[7]Aronsson, G., Evans, L. C. and Wu, Y., “Fast/slow diffusion and growing sandpiles”, J. Differential Equations 131 (1996) 304–335.Google Scholar
[8]Baiocchi, C., Buttazzo, G., Gastaldi, F. and Tomarelli, F., “General existence theorems for unilateral problems in continuum mechanics”, Arch. Rational Mech. Anal. 100 (1988) 149–189.Google Scholar
[9]Baiocchi, C. and Capelo, A., Variational and quasivariational inequalities: applications to free boundary problems (Wiley, New York, 1984).Google Scholar
[10]Brézis, H., “Equations et inéquations non linéaires dans les espaces vectoriels en dualité”, Ann. Inst. Fourier 18 (1968) 115–175.CrossRefGoogle Scholar
[11]Brézis, H., “Problèmes unilatéraux”, J. Math. Pures Appl. 51 (1972) 1–168.Google Scholar
[12]Cirmi, G. R. and Porzio, M. M., “L∞-solutions for some nonlinear degenerate elliptic and parabolic equations”, Ann. Mat. Pura Appl. 169 (1995) 67–86.Google Scholar
[13]Diaz, J. I., Nonlinear partial differential equations and free boundaries. Vol. 1: Elliptic equations, Research Notes in Math. 106 (Longman, London, 1985).Google Scholar
[14]Drabek, P., Kufner, A. and Nicolosi, F., “On the solvability of degenerated quasilinear elliptic equations of higher order”, J. Diff. Equations 109 (1994) 325–347.Google Scholar
[15]Drabek, P., Kufner, A. and Nicolosi, F., Quasilinear elliptic equations with degenerations and singularities (Walter de Gruyter, Berlin, 1997).Google Scholar
[16]Drabek, P. and Nicolosi, F., “Existence of bounded solutions for some degenerated quasilinear elliptic equations”, Ann. Mat. Pura Appl. 165 (1993) 217–238.Google Scholar
[17]Drabek, P. and Nicolosi, F., “Solvability of degenerate elliptic problems of higher order via Leray-Lions theorem”, Hiroshima Math. J. 26 (1996) 79–90.CrossRefGoogle Scholar
[18]Duvaut, G. and Lions, J. L., Les inéquations en mécanique et en physique (Dunod, Paris, 1972).Google Scholar
[19]Fonda, A. and Gossez, J.-P., “Semicoercive variational problems at resonance: an abstract approach”, Differential Integral Equations 3 (1990) 695–708.Google Scholar
[20]Heinonen, J., Kilpeläinen, T. and Martio, O., Nonlinear potential theory for degenerate elliptic equations (Cambridge Univ. Press, Cambridge, 1993).Google Scholar
[21]Hess, P., “On the solvability of nonlinear elliptic boundary value problems”, Indiana Univ. Math. J. 25 (1976) 461–466.CrossRefGoogle Scholar
[22]Huang, Y., “Existence of positive solutions for a class of the p-Laplace equations”, J. Austral. Math. Soc. Ser. B 36 (1994) 249–264.CrossRefGoogle Scholar
[23]Ivanov, A. V. and Mkrtycjan, P. Z., “On the solvability of the first boundary value problem for certain classes of degenerating quasilinear elliptic equations of the second order”, Proc. Steklov Inst. Math. 147 (1981) 11–35.Google Scholar
[24]Ward, J. R.Mawhin, J. and Willem, M., “Variational methods and semilinear elliptic equations”, Arch. Rational Mech. Anal. 95 (1986) 269–277.Google Scholar
[25]Kinderlehrer, D. and Stampacchia, G., An introduction to variational inequalities and their applications (Academic Press, New York, 1980).Google Scholar
[26]Krasnosels'kii, M. A., Topological methods in the theory of nonlinear integral equations (Pergamon Press, Oxford, 1963).Google Scholar
[27]Landesman, E. M. and Lazer, A. C., “Nonlinear perturbations of linear elliptic boundary value problems at resonance”, J. Math. Mech. 19 (1969/1970) 609–623.Google Scholar
[28]Le, V. K., “Some existence results for noncoercive nonconvex minimization problems with fast or slow perturbing terms”, Num. Func. Anal. Optim. 20 (1999) 37–58.CrossRefGoogle Scholar
[29]Le, V. K. and Schmitt, K., “Minimization problems for noncoercive functionals subject to constraints”, Trans. Amer. Math. Soc. 347 (1995) 4485–4513.CrossRefGoogle Scholar
[30]Le, V. K. and Schmitt, K., “Minimization problems for noncoercive functionals subject to constraints, Part II”, Adv. Differential Equations 1 (1996) 453–498.Google Scholar
[31]Le, V. K. and Schmitt, K., “On boundary value problems for degenerate quasilinear elliptic equations and inequalities”, J. Differential Equations 144 (1998) 170–218.Google Scholar
[32]Leonardi, S., “Solvability of degenerate quasilinear elliptic equations”, Nonlinear Analysis. TMA 26 (1996) 1053–1060.Google Scholar
[33]Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (Dunod, Paris, 1969).Google Scholar
[34]McKenna, P. J. and Rauch, J., “Strongly nonlinear perturbations of nonnegative boundary value problems with kernel”, J. Differential Equations 28 (1978) 253–265.Google Scholar
[35]Murthy, M. K. V. and Stampacchia, G., “Boundary value problems for some degenerate elliptic operators”, Ann. Math. Pura Appl. 80 (1968) 1–122.Google Scholar
[36]Opic, B. and Kufner, A., Hardy-type inequalities, Pitman Research Notes in Math. 219 (Longman, Harlow, 1990).Google Scholar
[37]Ôtani, M. and Teshima, T., “On the first eigenvalue of some quasilinear elliptic equations”, Proc. Japan Acad. 64 (1988) 8–10.Google Scholar
[38]Prigozhin, L., “Quasivariational inequality describing the shape of a poured pile”, Vychisl. Mat. Mat. Fiz. 26 (1986) 1072–1080.Google Scholar
[39]Prigozhin, L., “Sandpiles and river networks: extended systems with nonlocal interactions”, Phys. Rev. E 49 (1994) 1161–1167.Google Scholar
[40]Prigozhin, L., “Variational model of sandpile growth”, European J. Appl. Math. 7 (1996) 225–235.Google Scholar
[41]Rockafellar, R. T., “Level sets and continuity of conjugate convex functions”, Trans. Amer. Math. Soc. 123 (1966) 46–63.Google Scholar
[42]Rockafellar, T. R., Convex analysis (Princeton University Press, Princeton, NJ, 1970).Google Scholar
[43]Trudinger, N.. “Linear elliptic operators with measurable coefficients”, Ann. Scuola Norm. Sup. Pisa 27 (1973) 265–308.Google Scholar