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

Abstract

We consider existence and uniqueness of homogeneous solutions $u>0$ to certain PDE of $p$-Laplace type, $p$ fixed, $n-1<p<\infty ,n\ge 2,$ when $u$ is a solution in $K(\alpha )\subset {\mathbb{R}}^n$ where \[ K (α) := \{ x = (x_1, \dots, x_n ): x_1 > \cos α\, | x| \} \quad \mbox{for fixed}\, \, α\in (0, π], \] 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 $n=2,3$.

Incorporated referee comments: three figures and closing remarks added