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Hua Luo, Guowei Dai. BIFURCATION, A-PRIORI BOUND AND NEGATIVE SOLUTIONS FOR THE COMPLEX HESSIAN EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 937-963. doi: 10.11948/20200120

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Hua Luo, Guowei Dai. BIFURCATION, A-PRIORI BOUND AND NEGATIVE SOLUTIONS FOR THE COMPLEX HESSIAN EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 937-963. doi: 10.11948/20200120

BIFURCATION, A-PRIORI BOUND AND NEGATIVE SOLUTIONS FOR THE COMPLEX HESSIAN EQUATION

Hua Luo¹,
Guowei Dai^2, ,

1.
School of Economics and Finance, Shanghai International Studies University, Shanghai, 201620, China
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

Corresponding author: Email: daiguowei@dlut.edu.cn (G. Dai)
Fund Project: The authors were supported by NNSF of China (11871129), NSF of Liaoning Province (2019-MS-109) and HSSF of Chinese Ministry of Education (20YJA790049)

Abstract

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, a-priori bound of radially symmetric negative solutions is also obtained.
- Bifurcation /
- a-priori bound /
- complex Hessian equation
MSC: 35B32, 35B45, 35J60, 35P30, 53C56

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