The theory nowadays known as Brunn-Minkowski theory concerns the study of geometric functionals defined from convex bodies. A Minkowski problem is a characterization problem for a geometric measure generated by convex bodies. Its solution amounts to solving a degenerate fully nonlinear partial differential equation, see for instance [Y. Huang et al., Acta Math. 216, No. 2, 325–388 (2016; Zbl 1372.52007)].
Let \(\mathcal M_0\) be a smooth closed uniformly convex hypersurface in \(\mathbb R^{n+1}\) enclosing the origin. Consider the Gauss curvature flow \[ \begin{cases} \partial_t X(x,t)=-f(v)r^{\alpha}K(x,t)v,\\ X(x,0)=X_0(x), \end{cases} \] where

1.

\(K(\cdot,t)\) is the Gauss curvature of the hypersurface \(\mathcal M_t\) parameterized by \(X(\cdot,t):\mathbb S^{n}\rightarrow \mathbb R^{n+1} \),

2.

\(v(\cdot,t)\) is the unit outer normal vector at \(X(\cdot,t)\),

3.

\(f\) is a given smooth positive function on \(\mathbb{S}^n\),

4.

\(r=|X(x,t)|\) is the distance of \(X(x,t)\) to the origin.

The flow above was introduced to study the existence of solutions for the dual Minkowski problem proposed in [Y. Huang et al., loc. cit.]. This turns out to be equivalent to the following Monge-Ampère problem on \(\mathbb{S}^n\): \[ \mathrm{det}\left(\nabla^2u+uI\right)=\frac{f(x)}{u}\left(|\nabla u(x)|^2{+}u^2\right)^{\alpha/2}, \] where \(u\) denotes the support function of a hypersurface solution \(\mathcal{M}\).
The \(L_{p}\)-Minkowski problem, herein LpM, introduced in [E. Lutwak, J. Differ. Geom. 38, No. 1, 131–150 (1993; Zbl 0788.52007)], concerns the existence of closed convex hypersurfaces with prescribed \(p\)-area measure. By adapting the former Monge-Ampère equation, varying parameters include the dual problem to LpM. That is, \[ \mathrm{det}\left(\nabla^2u+uI\right)=\frac{f(x)u^{p-1}}{g\left(\frac{\nabla u(x)+ux}{\sqrt{|\nabla u(x)|^2+u(x)^2}} \right)}\left(|\nabla u(x)|^2{+}u^2\right)^{(n+1-q)/2}. \] This contains the dual LpM problem for \(g\equiv 1\).
The main result in the present paper provides sufficient existence conditions for this adapted Monge-Ampère problem assuming \(\alpha \in (0,1)\) and some regularity assumptions on \(f, g\) under constraints on \(p,q\).
The proof of the main result is achieved using the Gauss flow given above.