Positive solutions for a class of nonlinear singular boundary value problems at resonance. (English) Zbl 0805.34019

Bobisud, L. E.; O’Regan, Donal

J. Math. Anal. Appl. 184, No. 2, 263-284 (1994).

The paper deals with the singular differential equation \[ {1 \over p(t)} (p(t)y'(t))' = q(t) f(t,y(t), p(t),y'(t)),\quad t \in(0,1), \tag{1} \] with homogeneous two-point boundary conditions (of Dirichlet, Neumann, Robin or periodic type). Supposing the existence of upper and lower solutions and the growth conditions of the Bernstein-Nagumo type for \(f\), the authors prove the existence of a solution of the above boundary value problems. As a corollary they get the existence of a nonnegative solution. Further they study the singular Dirichlet problem (2) \({1 \over p(t)} (p(t)y'(t)' + \mu_ 0y(t) = q(t) f(t,y(t), p(t),y'(t))\), \(0<t<1\), \(y(0) = y(1)=0\), where \(\mu_ 0>0\) is the smallest eigenvalue of the associated linear problem. Two theorems which guarantee the existence of a nonnegative solution of (2) are presented here. The proofs are based on the upper and lower solutions method, the coincidence degree arguments and the topological transversality method.

Reviewer: I.Rachůnková (Olomouc)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

singular differential equation; homogeneous two-point boundary conditions; upper and lower solutions; growth conditions of the Bernstein-Nagumo type; existence; boundary value problems; nonnegative solution; singular Dirichlet problem; coincidence degree; topological transversality method