Twin solutions to singular Dirichlet problems. (English) Zbl 0946.34022

The authors consider the Dirichlet second-order boundary value problem \[ y''+ \phi(t)[g(y(t))+ h(y(t))]= 0,\quad 0< t< 1,\quad y(0)= y(1)= 0,\tag{1} \] and establish the existence of two solutions \(y_1,y_2\in C[0,1]\cap C^2(0, 1)\) with \(y_1> 0\), \(y_2> 0\) on \((0,1)\). The nonlinearity in (1) may be singular at \(y= 0\), \(t= 0\) and/or \(t= 1\).


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Agarwal, R.P.; O’Regan, D., Nonlinear superlinear singular and nonsingular second order boundary value problems, J. differential equations, 143, 60-95, (1998) · Zbl 0902.34015
[2] R. P. Agarwal, and, D. O’Regan, Twin solutions to singular boundary value problems, Proc. Amer. Math. Soc, to appear.
[3] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag New York · Zbl 0559.47040
[4] Eloe, P.W.; Henderson, J., Singular nonlinear boundary value problems for higher order ordinary differential equations, Nonlinear anal., 17, 1-10, (1991) · Zbl 0731.34015
[5] Eloe, P.W.; Henderson, J., Inequalities based on a generalization of concavity, Proc. amer. math. soc., 125, 2103-2107, (1997) · Zbl 0868.34008
[6] Erbe, L.H.; Hu, S.; Wang, H., Multiple positive solutions of some boundary value problems, J. math. anal. appl., 184, 640-648, (1994) · Zbl 0805.34021
[7] O’Regan, D., Existence of nonnegative solutions to superlinear non-positone problems via a fixed point theorem in cones of Banach spaces, Dynamics of continuous, discrete and impulsive systems, 3, 517-530, (1997) · Zbl 0909.34017
[8] O’Regan, D., Existence theory for nonlinear ordinary differential equations, (1997), Kluwer Dordrecht · Zbl 1077.34505
[9] O’Regan, D., Existence principles and theory for singular Dirichlet boundary value problems, Differential equations dynam. systems, 3, 289-304, (1995) · Zbl 0876.34018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.