Bifurcation, apriori bound and negative solutions for the complex Hessian equation 
Hua Luo,Guowei Dai 
Keywords:Bifurcation; Apriori bound; Complex Hessian equation 
Abstract: 
This paper establishes global bifurcation and eigenvalue results for the following complex $k$Hessian equation
\begin{equation}
\left\{
\begin{array}{ll}
S_k\left(u_{i\overline{j}}\right)=\lambda^k f(u)\,\, &\text{in}\,\, B,\u=0 &\text{on}\,\, \partial B.
\end{array}
\right.\nonumber
\end{equation}
The existence/nonexistence, uniqueness and multiplicity of radially symmetric negative solutions are investigated. Moreover, apriori bound of radially symmetric negative solutions is also obtained. 



