Multiplicity of positive solutions for an equation with degenerate nonlocal diffusion

Even without a variational background, a multiplicity result of positive solutions with ordered $L^p(\mathrm{\Omega })$-norms is provided to the following boundary value problem \begin{equation*} \left { \begin{array}{ll} -a(\int_{\Omega}u^{p}dx)\Delta u = f(u) & \mbox{in $\mathrm{\Omega }$,}\ u=0 & \mbox{on $\mathrm{\partial }\mathrm{\Omega }$,} \end{array}\right. \end{equation*} where $\mathrm{\Omega }$ is a bounded domain and $a$, $f$ are continuous real functions with $a$ vanishing in many positive points.