Let \(s^{(p)}=| s| ^{p-1}s\) (\(s\) real). The differential operator \[ L_p^\alpha=r^{-\alpha}\bigl(r^\alpha{u'}^{p-1}\bigr)' \] is considered, where \(s\) is the independent variable, \(\alpha\geq 0\), and \(p>1\). For \(\alpha=n-1\) and \(r=| x| \), this is the radial \(\Delta_p\)-operator in \(\mathbb{R}^n\). It is shown that the initial value problem \[ L_p^\alpha u+q(r)u^{(p-1)}=0,\quad u(r_0)=0,\quad u'(r_0)=u_0',\quad (u_0'=0\text{ if }r_0=0), \] has a unique solution, where \(q\) is continuous. It is noted that the uniqueness is not true in general for the corresponding inhomogeneous problem. Using the Prüfer transformation, the following comparison theorem of Sturmian type is shown:
Let \(u_1\), \(u_2\) be nontrivial solutions to \(L_p^\alpha u_jq_j(r)u_j^{(p-1)}=0\), \(j=1,2\), where \(q_1\leq q_2\).
(a) If \(I=[0,b]\), \(q_1\not\equiv q_2\), \(u_1'(0)=0\), \(u_1(b)=0\), and \(u_2'(0)=0\), then \(u_2\) has a zero in \((0,b)\).
(b) If \(I=[a,b]\), \(a>0\), \(u_1(a)=u_1(b)=0\), then either (i) \(u_2=\lambda u_1\) or (ii) \(u_2\) has a zero in \((a,b)\). It is also shown that the eigenvalue problem \[ L_p^\alpha u+(q(r)+\lambda s(r))u^{(p-1)}=0,\;u'(0)=0,\;u(b)=0, \] on \([0,b]\) with \(s\) positive has a countable number of real eigenvalues with limit point \(\infty\). The location of the zeros of the corresponding eigenfunctions is investigated.