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Existence of positive solutions for a class of the p-Laplace equations

Published online by Cambridge University Press:  17 February 2009

Yin Xi Huang
Yin Xi Huang
Affiliation:
Department of Mathematical Sciences, Memphis State University, Memphis, TN 38152, U.S.A
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Abstract

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We are concerned with the existence of solutions of

where Δp is the p-Laplacian, p ∈ (1, ∞), and Ω is a bounded smooth domain in ℝn.

For h(x) ≡ 0 and f(x, u) satisfying proper asymptotic spectral conditions, existence of a unique positive solution is obtained by invoking the sub-supersolution technique and the spectral method. For h(x) ≢ 0, with assumptions on asymptotic behavior of f(x, u) as u → ±∞, an existence result is also proved.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 2 , October 1994 , pp. 249 - 264
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Anane, A., “Simplicité et isolation de la première valeur propre du p-laplacien avec poids”, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 725–728.Google Scholar
[2]Azorero, J. P. G. and Alonso, I. P., “Existence and uniqueness for the p-Laplacian: nonlinear eigenvalues”, Comm. Partial Differential Equations 12 (1987) 1389–1430.Google Scholar
[3]Boccardo, L., Drábek, P. and Kučera, M., “Landesman-Lazer conditions for strongly nonlinear boundary value problems”, Comment. Math. Univ. Carolin. 30 (1989) 411–427.Google Scholar
[4]Brezis, H. and Oswald, L., “Remarks on sublinear elliptic equations”, Nonlinear Anal. 10 (1986) 55–64.CrossRefGoogle Scholar
[5]Costa, D. G. and Gonçalves, J. V. A., “On the existence of positive solutions for a class of non-selfadjoint elliptic boundary value problems”, Applicable Anal. 31 (1989) 309–320.CrossRefGoogle Scholar
[6]Pino, M. Del, Elgueta, M. and Manasevich, R., “A homotopic deformation along p of a Leray-Schauder degree result and existence for (|u′|p-2u′)′ + f(t, u) = 0, u(0) = u(t) = 0, p 1”, J. Differential Equations 80 (1989) 1–13.CrossRefGoogle Scholar
[7]Benedetto, E. Di, “C 1+α local regularity of weak solutions of degenerate elliptic equations”, Nonlinear Anal. 7 (1983) 827–850.CrossRefGoogle Scholar
[8]Diaz, J. I., Nonlinear Partial Differential Equations and Free Boundaries vol. I: Elliptic Equations, Volume 106 of Research Notes in Mathematics (Pitman Advanced Publishing Program, London, 1985).Google Scholar
[9]Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed. (Springer-Verlag, New York, 1983).Google Scholar
[10]Guedda, M. and Veron, L., “Local and global properties of solutions of quasilinear elliptic equations”, J. Differential Equations 76 (1988) 159–189.CrossRefGoogle Scholar
[11]Hess, P. and Kato, T., “On some linear and nonlinear eigenvalue problems with an indefinite weight function”, Comm. Partial Differential Equations 5 (1980) 999–1030.CrossRefGoogle Scholar
[12]Kazdan, J. and Kramer, R. J., “Invariant criteris for existence of solutions to second order quasilinear equations”, Comm. Pure Appl. Anal. 31 (1978) 619–645.Google Scholar
[13]Lindqvist, P., “On the equation div (|∇u|p−2∇) + γ|u|p−2u”, Proc. Amer. Math. Soc. 109 (1990) 157–164.Google Scholar
[14]Lions, J. L., Quelques Méthodes de Résolution des Problémes aux Limites Non-linéaires (Dunod, Paris, 1969).Google Scholar
[15]Otani, M., “Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations”, J. Funct. Anal. 76 (1988) 140–159.CrossRefGoogle Scholar
[16]Otani, M. and Teshima, T., “On the first eigenvalue of some quasilinear elliptic equations”, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988) 8–10.Google Scholar
[17]Sattinger, D. H., Topics in Stability Bifurcation Theory, Volume 309 of Lecture Notes in Mathematics (Springer-Verlag, New York, 1973).CrossRefGoogle Scholar
[18]Serrin, J., “Local behaviour of solutions of quasilinear equations”, Acta Math. 111 (1964) 247–302.CrossRefGoogle Scholar
[19]Szulkin, A., “Ljusternik-Schnirelmann theory on C 1 manifolds”, Ann. Inst. H. Poincaré. Anal. Non Linéaire 5 (1988) 119–139.CrossRefGoogle Scholar
[20]Tolksdorf, P., “On the Dirichlet problem for quasilinear equations in domains with conical boundary points”, Comm. Partial Differential Equations 8 (1983) 773–817.CrossRefGoogle Scholar