Bernstein properties of solutions to some higher-order equations

Abstract

We record some Bernstein properties for entire solutions to higher-order elliptic equations, including certain bi-Laplacian equations, other equations which are second order in the eigenvalues of the Hessian of the solution and some geometrically inspired fourth-order equations. We apply a Bernstein result for second-order elliptic equations on noncompact Riemannian manifolds, whose proof is based on techniques of Yau [20].