For even prescribed functions satisfying explicit curvature conditions, the horospherical p-Christoffel-Minkowski equation σ_k(A[φ]) = φ^{p-k} f admits a smooth uniformly h-convex solution φ > 1 on S^n.
Caffarelli and Avner Friedman, Convexity of solutions of semilinear elliptic equations , Duke Math
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The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space
For even prescribed functions satisfying explicit curvature conditions, the horospherical p-Christoffel-Minkowski equation σ_k(A[φ]) = φ^{p-k} f admits a smooth uniformly h-convex solution φ > 1 on S^n.