Interior estimates for solutions of a fourth order nonlinear partial differential equation (Q2468461)

The authors establish interior estimates for solutions of the fourth-order nonlinear PDE \[ \Delta\rho=\alpha{{| \nabla\rho| ^2_G}\over{\rho}},\tag{\(*\)} \] which can be applied to uniqueness questions for solutions of (\(*\)). It is proved that a Euclidean complete \(\alpha\)-extremal hypersurface must be an elliptic paraboloid for \(| \alpha| \geq K(n)\).