Boundary asymptotical behavior of large solutions to complex Hessian equations
The author considers the following fully nonlinear singular Dirichlet problem for symmetric functions of the complex Hessian over bounded smooth domains \(\Omega \subset \mathbb C^n\) \[ \sigma_k\left(\frac {\partial^2 u}{\partial z_i \partial \overline z_j}\right)=g(z)f(u) \;\text{in} \;\Omega \subset \mathbb C^n,\quad u=+\infty \;\text{on} \;\partial \Omega, \] for \(k\leq n\), where \(\sigma_k\) is the \(k\)-th symmetric polynomial of the eigenvalue of the complex Hessian, and \(u\) is strictly \(\Gamma_k\)-subharmonic. (A \(C^2\) function \(u\) is strictly \(\Gamma_k\)-subharmonic if \(\sigma_j(\frac {\partial^2 u}{\partial z_i \partial \overline z_j})>0\) for \(j=1,2,\dots ,k\).) Under the assumption that \(f\) satisfies \(\lim_{t\to \infty} \frac {f(\lambda t)}{f(t)} =\lambda^q\) for \(q>n\), and \(g(z)\geq 0 \) and \(\frac {g(z)}{m^{k+1}(dz)}\) are bounded from above and below by some positive constant as \(z\to \partial \Omega\) (where \(m\) is some nondecreasing function on \((0,\mu)\) such that \(\lim_{t\to 0}\frac d{dt}(\int_0^tm(s)ds/m(t))=l\)), the author establishes the asymptotical behavior of any \(C^2\) solution to the above problem. The statement of the main result and the techniques of the proof are quite similar to the work on singular Dirichlet problems for real Monge-Ampère equation by \textit{F. C. Cîrstea} and \textit{C. Trombetti} [Calc. Var. Partial Differ. Equ. 31, No. 2, 167--186 (2008; Zbl 1148.35022)].
scientific article; zbMATH DE number 5817665
scientific article