Positive radial solutions involving nonlinearities with zeros

In this paper we consider the non-autonomous quasilinear elliptic problem $$ \begin{cases} -\Delta_p u=\lambda |x|^{\delta} f(u) &\mbox{in }B_1(0)\\ u=0 &\mbox{in }\partial B_1(0), \end{cases} $$ where $f:\mathbb{R}\to[0,\infty)$ is a nonnegative $C^1-$function with $f(0)=0$, $f(U)=0$ for some $U>0$, and $f$ is superlinear in $0$ and in $U$. Assuming subcriticality either in $U$ or at infinity we study existence and multiplicity of positive radial solutions with respect to the parameter $\lambda$. In addition, we study the bifurcation diagrams with respect to the maximum over the eventual solutions as the parameter $\lambda$ varies in the positive halfline.