Positive Solutions of Semilinear Differential Equations with Singularities

✍ Scribed by Kunquan Lan; Jeffrey R.L. Webb

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 266 KB
• Volume: 148
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1998.3475

✦ Synopsis

The existence of positive solutions of a second order differential equation of the form z"+ g(t) f (z)=0

(1.1) with suitable boundary conditions has proved to be important in theory and applications whether g is continuous in [0, 1] or g has singularities.

These equations often arise in the study of positive radial solutions of a nonlinear elliptic equation of the form

for example, see [3, 7, 8, 18, and 23]. Moreover, Eq. (1.1) contains many important equations which arise from other fields. For example, the generalized Emden Fowler equation, where f =z p , p>0 and g is continuous (see [21] and [24]), arises in the fields of gas dynamics, nuclear physics, and chemically reacting systems [24]; and the Thomas Fermi equation, where f =z 3Â2 and g=t &1Â2 , so g has a singularity at 0 (see [9, 10 and 21]), was developed in studies of atomic structures (see, for example, [21]) and atomic calculations [5]. When g is continuous, the existence of positive solutions of Eq. (1.1) with suitable boundary conditions has been studied in [23] by using normtype cone expansion and compression theorems. The key conditions on f are either f is superlinear, that is, lim x Ä 0 f (x)Âx=0 and lim x Ä f (x)Âx= or f is sublinear, that is, lim x Ä 0 f (x)Âx= and lim x Ä 0 f (x)Âx=0. However, it is known that Eq. (1.1) with g#1 has positive solutions for article no. DE983475