Closed convex surfaces in $E^3$ with given functions of curvatures

It is proved that there are a regular closed convex surface $S$ and a constant vector $c$ for which the equality
$$K^{-1}+H^{-\alpha}+c\mathbf n=\varphi(\mathbf n)$$
is realized at a point with external normal $\mathbf n$. Here $K$ and $H$ are the Gauss and mean curvatures of $S$ at the point with normal $\mathbf n$, $\varphi(\mathbf n)$ is a given regular function on sphere, which satisfies the closeness condition and the inequality
$$\operatorname{inf}\varphi>\frac9{32}\biggl[1+\sqrt{1+\frac{64}9(\operatorname{sup}\varphi)^{2-\alpha}}\biggr](\operatorname{sup}\varphi)^{\alpha-1},$$
$\alpha\in(0,1]$. The solution $(S,c)$ is unique with a translation.