Positive solutions of a 2nd order singular ordinary differential equation
by
J.M. Gomes
CMAF, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa , Portugal
Coauthors: D. Bonheure and L. Sanchez

Abstract

u''+k  u' t =c(t)g(u)         (k > 1)\tag*(1)

u'(0)=0,     u(M)=0,       (0 < M <= \infty)

As is well known, when k=N-1, this problem concerns the existence of radial solutions to the elliptic equation

\Deltau=c(|x|)g(u)

in a ball or in R^N. We assume 0 < c₁ <= c(t) <= c₂, g is bounded, g(0)=0,     g(u)(a-u) > 0  for u =/= a, and \int₀^\xi g(u)du < 0 for a certain \xi > 0.

 In a paper of 1981, Berestycki, Lions and Peletier consider a version of this problem with a non-autonomous nonlinearity with monotone dependence on t. Althought their purpose is an ODE approach to problem (1), a certain step of the proof needs the solution of a variational elliptic problem in a ball. This has motivated us to look for a strict ODE approach where in particular k could be taken to be a real number, not necessarily an integer.

 Combining variational methods in appropriate weighted Sobolev spaces with the method of non-well ordered lower and upper solutions we obtain the following result under the above prescribed assumptions:

 Take [`a] to be such that \int<sub>0</sub><sup>[`a]</sup>g(u)du=0. Then exists M₀ > 0 with the following properties: the problem (1) for finite M has:

(i) at least two positive solutions if M > M₀;

(ii) at least one positive solution with initial value >= [`a] \ if M=M₀;

(iii) no positive solution with initial value >= [`a] if M < M₀.

The case M=\infty is handled with a topological shooting argument similarto the one used by the authors mentioned above.

Date received: May 20, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-53.