A Pohožaev Identity and Critical Exponents of Some Complex Hessian Equations

Abstract

In this paper, we prove some sharp non-existence results for Dirichlet problems of complex Hessian equations. In particular, we consider a complex Monge-Ampère equation which is a local version of the equation of Kähler-Einstein metric. The non-existence results are proved using the Pohožaev method. We also prove existence results for radially symmetric solutions. Themain difference of the complex case with the real case is that we don't know if a priori radially symmetric property holds in the complex case.