Multiple sign changing solutions in a class of quasilinear equations. (English) Zbl 1020.34019

Goncalves, J. V.; Melo, A. L.

Differ. Integral Equ. 15, No. 2, 147-165 (2002).

The authors consider the boundary value problem \[ -(r^\alpha|u'(r)|^\beta u'(r))'= \lambda r^\gamma f(u(r)),\quad 0< r< R,\quad u(R)= u'(0)= 0, \] where \(\alpha\), \(\beta\), \(\gamma\) are given numbers, \(\lambda> 0\) is a parameter, the function \(f\) is continuous and \(tf(t)> 0\) for \(t\neq 0\). Conditions are obtained that, for all sufficiently small \(\lambda\), guarantee the existence of a sequence \(u_l\) of classical solutions such that the function \(u_l\) has precisely \(l\) zeroes in \((0,R)\). The proof employs an appropriate family of initial value problems and is based on the shooting technique.

Reviewer: Sergei A.Brykalov (Ekaterinburg)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
35J25	Boundary value problems for second-order elliptic equations

Keywords:

nonuniqueness of solutions; number of zeroes; sign-changing solutions