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