Bounded positive solutions of rotationally symmetric harmonic map equations. (English) Zbl 0984.34018

Cheung, Leung-Fu; Law, Chun-Kong; Leung, Man-Chun

Differ. Integral Equ. 13, No. 7-9, 1149-1188 (2000).

The authors study the second-order differential equation \[ \alpha''(r)+(n-1){{f'(r)}\over{f(r)}}\alpha'(r)- (n-1){{g(\alpha(r))g'(\alpha(r))}\over{f^2(r)}}=0 \] with the initial limit \(\alpha(0)=0\). Here, the real functions \(f\) and \(g\) are positive on \((0,\infty)\), etc. First, they discuss existence and uniqueness of positive solutions \(\alpha\) with prescribed \(\alpha'(0)\) as well as the behaviour of the map \(\alpha'(0)\to \alpha(1)\). Further, they identify conditions on \(f\) and \(g\) for the following four possible cases to occur: all saturated, positive solutions are defined on bounded domains where they are unbounded; all saturated, positive solutions defined on \((0,\infty)\) are unbounded; some saturated, positive solutions defined on \((0,\infty)\) are bounded; all saturated, positive solutions defined on \((0,\infty)\) are bounded. These conditions involve the improper integrals \(\int_1^\infty{{dr}\over{f(r)}}\) and \(\int_1^\infty{{dr}\over{g(r)}}\). In the last case, the behaviour of the map \(\alpha(1)\to\alpha(\infty)\) is studied.

Reviewer: Corneliu Ursescu (Iaşi)

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
58E20	Harmonic maps, etc.

Keywords:

bounded positive solutions; existence; uniqueness; behaviour