A Priori Estimate Of The Solution Of A Class Of Fully Nonlinear σ₂-Yamabe Equations

The famous Yamabe problem is about a class of nonlinear partial differential equations in conformal geometry.Precisely speaking,the Yamabe problem reads:given a smooth,compact Riemannian manifold（M,g）without boundary of dimensional n≥3,does there exist a metric g conformal to g for which scalar curvature of g is a constant?σk-Yamabe problem is a generalization of Yamabe problem.On the basis of the classic Yamabe problem,which generalize the constant scalar curvature to the constant σk-curvature of Schouten tensor,where σk-curvature refers to the k-th symmetric polynomial of the eigenvalue for the second order symmetric Schouten tensor.In case k=1,σ1-curvature is scalar curvature and σ1-Yamabe problem corresponds to the classical Yamabe problem.The further studies on σkYamabe problem is of great significance to promote the development of fully nonlinear partial differential equations.In this paper,we consider the priori estimates of the solution of a class of fully nonlinear σ2-Yamabe equation with 0 Dirichlet boundary value conditions.It generalize the Brezis-Nirenberg problem about semilinear Yamabe equation of Laplace operator to the fully nonlinear σ2-Yamabe equation.The gradient estimates to the solution of the σ2-Yamabe equation is studied by constructing an auxiliary function.First,we reduced the interior gradient estimates to the the boundary.And then the boundary gradient estimates is given by the method of barrier function.Finally,the global gradient estimates of the solution is obtained.