Let y(t) be a solution of the equation \(q(t,y'))'+a(t)f(y)=0\). In this paper, questions about the qualitative behaviour, such as the positivity, or the oscillatory character, of y on a half-line (T,\(\infty)\) are discussed in relation to structure hypotheses on the functions q,a and f. Particular examples are: \(q(t,v)=| v|^{m-2}v\) and \(q(t,v)=v(1+t^ kv^ 2)^{-},\) \(a(t)=t^{-k}\) and \(f(y)=-y+| y|^{p-1}y,\) where m,k and p are appropriately related. These results yield as corollary nonexistence theorems for radially symmetric ground state solutions of quasi-linear elliptic equations of the form \(div(A(| Du|)Du)+f(u)=0,\) of the type recently obtained by \textit{W. M. Ni} and \textit{J. Serrin} [``Non-existence theorems for quasilinear partial differential equations'', Rend. Circ. Mat. Palermo, Suppl. Studies (to appear); Commun. Pure Appl. Math. 39, 379-399 (1986; Zbl 0602.35041)].