Uniqueness of positive solutions of quasilinear differential equations. (English) Zbl 0831.34028

Naito, Yūki

Differ. Integral Equ. 8, No. 7, 1813-1822 (1995).

The paper deals with differential equations of the form (1) \(|u' |m - 2u')' + p(t) f(u) = 0\), \(t \in (a,b)\), where \(m > 1\), \(p\) is continuous and positive on \((a,b)\), \(f\) is continuous on \(0, \infty)\), positive on \((0, \infty)\) and such that the function \({f(u) \over um - 1}\) is decreasing in \(u \in (0, \infty)\). In particular, the author proves uniqueness theorems for positive solutions of the equation (1) subjected to the boundary conditions \(u(a) = u'(b) = 0\), \(u'(a) = u(b) = 0\) and \(u(a) = u(b) = 0\), respectively. To this aim the notion of a generalized Prüfer transformation is introduced.

Reviewer: M.Tvrdý (Praha)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
47J05	Equations involving nonlinear operators (general)

Keywords:

boundary value problem; quasilinear differential equation; uniqueness; generalized Prüfer transformation