Uniqueness of critical points and maximum principles of the singular minimal surface equation

摘要

We consider a smooth solution u 0 of the singular minimal surface equation root 1 + vertical bar Du vertical bar(2) div(Du/root 1 + vertical bar Du vertical bar(2))=alpha/u defined in a bounded strictly convex domain of R-2 with constant boundary condition. If alpha 0, we prove the existence a unique critical point of u. We also derive some C-0 and C-1 estimates of u by using the theory of maximum principles of Payne and Philippin for a certain family of Phi-functions. Finally we deduce an existence theorem of the Dirichlet problem when alpha 0. (C) 2018 Elsevier Inc. All rights reserved.