Note on an eigenvalue problem with applications to a Minkowski type regularity problem in Rⁿ

We consider existence and uniqueness of homogeneous solutions u > 0 to certain PDE of p-Laplace type, p fixed, n - 1 < p <∞, n ≥ 2, when u is a solution in K(α) ⊂ Rⁿ where K(α) := { x = (x₁,..., x_n ) : x₁ > cos α | x| } for fixed α ∈ (0, π ], with continuous boundary value zero on ∂K (α) \ {0\}. In our main result we show that if u has continuous boundary value 0 on ∂K (π)$ 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 Rⁿ when n = 2; 3.