## 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.

### MSC:

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