A Pohozaev identity and critical exponents of some complex Hessian equations

TL;DR

This paper investigates complex Hessian equations, establishing non-existence results via Pohozaev identities and proving existence of radially symmetric solutions, highlighting differences from real cases.

Contribution

It introduces non-existence results for complex Hessian equations using Pohozaev identities and explores existence of radially symmetric solutions, addressing a key difference from real cases.

Findings

Non-existence results for certain Dirichlet problems.

Existence of radially symmetric solutions.

Highlighting differences between complex and real Hessian equations.

Abstract

In this note, we prove some non-existence results for Dirichlet problems of complex Hessian equations. The non-existence results are proved using the Pohozaev method. We also prove existence results for radially symmetric solutions. The main 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.