Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T13:08:23.342Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigenvalue characterization for (n · p) boundary-value problems

Published online by Cambridge University Press:  17 February 2009

Patricia J. Y. Wong  and
Ravi P. Agarwal
Patricia J. Y. Wong
Affiliation:
Division of Mathematics, Nanyang Technological University, 469 Bokit Timah Road, Singapore259756
Ravi P. Agarwal
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore119260
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the (n, p) boundary value problem

where λ > 0 and 0 ≤ p ≤ n - l is fixed. We characterize the values of λ such that the boundary value problem has a positive solution. For the special case λ = l, we also offer sufficient conditions for the existence of positive solutions of the boundary value problem.

Type
Research Article
Information
The ANZIAM Journal , Volume 39 , Issue 3 , January 1998 , pp. 386 - 407
Copyright
Copyright © Australian Mathematical Society 1998

References

[1] Agarwal, R. P., “Some new results on two-point problems for higher order differential equations”, Funkcialaj Ekvacioj 29 (1986) 197–212.Google Scholar
[2] Agarwal, R. P., Boundary value problems for higher order differential equations (World Scientific, Singapore, 1986).CrossRefGoogle Scholar
[3] Agarwal, R. P. and Henderson, J., “Superlinear and sublinear focal boundary value problems”, Applicable Analysis, (to appear).Google Scholar
[4] Agarwal, R. P. and Wong, P. J. Y., “Existence of solutions for singular boundary value problems for higher order differential equations”, Rendiconti del Seminario Matematico e Fisico di Milano 55 (1995) 249–264.CrossRefGoogle Scholar
[5] Aronson, D., Crandall, M. G. and Peletier, L. A., “Stabilization of solutions of a degenerate nonlinear diffusion problem”, Nonlinear Analysis 6 (1982) 1001–1022.CrossRefGoogle Scholar
[6] Bandle, C., Coffman, C. V. and Marcus, M., “Nonlinear elliptic problems in annular domains”, J. Diff. Eqns. 69 (1987) 322–345.CrossRefGoogle Scholar
[7] Bandle, C. and Kwong, M. K., “Semilinear elliptic problems in annular domains”, J. Appl. Math. Phys. 40 (1989) 245–257.Google Scholar
[8] Cac, N. P., Fink, A. M. and Gatica, J. A., “Nonnegative solutions of quasilinear elliptic boundary value problems with nonnegative coefficients”, preprint.Google Scholar
[9] Choi, Y. S. and Ludford, G. S., “An unexpected stability result of the near-extinction diffusion flame for non-unity Lewis numbers”, Q. J. Mech. Appl. Math. 42 (1989) 143–158.CrossRefGoogle Scholar
[10] Chyan, C. J. and Henderson, J., “Positive solutions for singular higher order nonlinear equations”, Diff. Eqs. Dyn. Sys. 2 (1994) 153–160.Google Scholar
[11] Cohen, D. S., “Multiple stable solutions of nonlinear boundary value problems arising in chemical reactor theory”, SIAM J. Appl. Math. 20 (1971) 1–13.CrossRefGoogle Scholar
[12] Dancer, E. N., “On the structure of solutions of an equation in catalysis theory when a parameter is large”, J. Diff. Eqns. 37 (1980) 404–437.CrossRefGoogle Scholar
[13] Eloe, P. W. and Henderson, J., “Singular nonlinear boundary value problems for higher order ordinary differential equations”, Nonlinear Analysis 17 (1991) 1–10.CrossRefGoogle Scholar
[14] Eloe, P. W. and Henderson, J., “Positive solutions for higher order ordinary differential equations”, Electronic J. of Differential Equations 3 (1995) 1–8.Google Scholar
[15] Eloe, P. W. and Henderson, J., “Positive solutions for (n – 1, 1) boundary value problems”, Nonlinear Analysis, (to appear).Google Scholar
[16] Eloe, P. W., Henderson, J. and Wong, P. J. Y., “Positive solutions for two-point boundary value problems”, in Dynamic Systems and Applications (eds. Ladde, G. S. and Sambandham, M.), Volume 2 (1996) 135–144.Google Scholar
[17] Erbe, L. H. and Wang, H., “On the existence of positive solutions of ordinary differential equations”, Proc. Amer. Math. Soc. 120 (1994) 743–748.CrossRefGoogle Scholar
[18] Fink, A. M., The radial Laplacian Gel'fund problem, Delay and Differential Equations, (Ames, IA, 1991), 93–98 (World Scientific, River Edge, NJ, 1992).Google Scholar
[19] Fink, A. M., Gatica, J. A. and Hernandez, E., “Eigenvalues of generalized Gel'fand models”, Nonlinear Analysis 20 (1993) 1453–1468.CrossRefGoogle Scholar
[20] Fink, A. M. and Gatica, J. A., “Positive solutions of systems of second order boundary value problems”, J. Math. Anal. Appl., (to appear).Google Scholar
[21] Fujita, H., “On the nonlinear equations and ”, Bull. Amer. Math. Soc. 75 (1969) 132–135.CrossRefGoogle Scholar
[22] Garaizar, X., “Existence of positive radial solutions for semilinear elliptic problems in the annulus”, J. Diff. Eqns. 70 (1987) 69–72.CrossRefGoogle Scholar
[23] Gatica, J. A., Oliker, V. and Waltman, P., “Singular nonlinear boundary value problems for second-order ordinary differential equations”, J. Diff. Eqns. 79 (1989) 62–78.CrossRefGoogle Scholar
[24] Gel'fand, I. M., “Some problems in the theory of quasilinear equations”, Uspehi Mat. Nauka 14 (1959) 87–158, English translation, Trans. Amer. Math. Soc.29 (1963), 295–381.Google Scholar
[25] Henderson, J., “Singular boundary value problems for difference equations”, Dynamic Systems and Applications 1 (1992) 271–282.Google Scholar
[26] Henderson, J., “Singular boundary value problems for higher order difference equations”, in Proceedings of the First World Congress on Nonlinear Analysis (ed. Lakshmikantham, V.), (Walter de Gruyter and Co., 1996).Google Scholar
[27] Krasnosel'skii, M. A., Positive Solutions of Operator Equations (Nordhoff, Groningen, 1964).Google Scholar
[28] O'Regan, D., Theory of Singular Boundary Value Problems (World Scientific, Singapore, 1994).CrossRefGoogle Scholar
[29] Parter, S., “Solutions of differential equations arising in chemical reactor processes”, SIAM J. Appl. Mart. 26 (1974) 687–716.CrossRefGoogle Scholar
[30] Wang, H., “On the existence of positive solutions for semilinear elliptic equations in the annulus”, J. Diff. Eqns. 109 (1994) 1–7.CrossRefGoogle Scholar
[31] Wong, P. J. Y. and Agarwal, R. P., “On the existence of solutions of singular boundary value problems for higher order difference equations”, Nonlinear Analysis 28 (1997) 277–287.CrossRefGoogle Scholar
[32] Wong, P. J. Y. and Agarwal, R. P., “On the existence of positive solutions of higher order difference equations”, Topological Methods in Nonlinear Analysis, (to appear).Google Scholar
[33] Wong, P. J. Y. and Agarwal, R. P., “On the eigenvalues of boundary value problems for higher order difference equations”, Rocky Mountain J. Math., (to appear).Google Scholar
[34] Wong, P. J. Y. and Agarwal, R. P., “Eigenvalues of boundary value problems for higher order differential equations”, Mathematical Problems in Engineering 2 (1996) 401–434.CrossRefGoogle Scholar