On compact anisotropic Weingarten hypersurfaces in Euclidean space

Let $$\Omega \subset \mathbb {R}^n$$ be a bounded mean convex domain. If $$\alpha <0$$ , we prove the existence and uniqueness of classical solutions of the Dirichlet problem in $$\Omega $$ for the $$\alpha $$ -singular minimal surface equation with arbitrary continuous boundary data.