  We consider a fully nonlinear partial differential equation associated to the
intermediate $L^p$ Christoffel-Minkowski problem in the case $1<p<k+1$. We
establish the existence of convex body with prescribed $k$-th even $p$-area
measure on $\mathbb S^n$, under an appropriate assumption on the prescribed
function. We construct examples to indicate certain geometric condition on the
prescribed function is needed for the existence of smooth strictly convex body.
We also obtain $C^{1,1}$ regularity estimates for admissible solutions of the
equation when $ p\ge \frac{k+1}2$.
