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

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.


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