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

$L^p$ Christoffel-Minkowski problem: the case $1< p