  The $L_p$-Minkowski problem deals with the existence of closed convex
hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It
extends the classical Minkowski problem and embraces several important
geometric and physical applications. The Existence of solutions has been
obtained in the sub-critical case $p>-n-1$, but the problem remains widely open
in the super-critical case $p<-n-1$. In this paper, we introduce new ideas to
solve the problem for all the super-critical exponents. A crucial ingredient in
our proof is a topological method based on the calculation of the homology of a
topological space of ellipsoids.
