Note on an eigenvalue problem with applications to a Minkowski type regularity problem in $\mathbb{R}$^n

We consider existence and uniqueness of homogeneous solutions $u > 0$ to certain PDE of $p$-Laplace type, $p$ fixed, $n - 1 <p< \infty, n \geq 2,$ when $u$ is a solution in $K(\alpha)\subset\mathbb{R}^n$ where $$K (\alpha) := \{ x = (x_1, \dots, x_n ): x_1 > \cos \alpha \, | x| \} \quad \mbox{for fixed}\, \, \alpha \in (0, \pi ],$$ with continuous boundary value zero on $\partial K ( \alpha ) \setminus \{0\}$. In our main result we show that if $u$ has continuous boundary value $0$ on $\partial K ( \pi )$ then $u$ is homogeneous of degree $1 - (n-1)/p$ when $p > n - 1.$ Applications of this result are given to a Minkowski type regularity problem in $\mathbb{R}^{n}$ when $n=2,3$.