Abstract
Based on the asymptotic property of the level hypersurfaces of f, we show that the solvability of Dirichlet problem for the fully nonlinear elliptic equation with \(\varGamma =\varGamma _n\) is closely related to the existence of a \(\mathcal {C}\)-subsolution introduced by Székelyhidi (J Differ Geom 109: 337–378, 2018) of a rescaled equation. For the complex Monge–Ampère equation and complex Hessian equations the gradient estimate established in previous works (Błocki, Math Ann 344: 317–327, 2009; Hanani, J Funct Anal 137: 49–75, 1996; Guan and Li, Adv Math 225: 1185–1223, 2010; Zhang, Int Math Res Not 2010: 3814–3836, 2010) also follows as a consequence of our argument. Also, the existence and regularity of admissible solutions to Dirichlet problem for fully nonlinear elliptic equations on compact Kähler manifolds of nonnegative orthogonal bisectional curvature are obtained.
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Notes
Following an idea of new version of [13], one can improve the part a) in (43) as follows:
$$\begin{aligned} \begin{aligned} \sum _{i=1}^n f_i |\lambda _i| \le \epsilon \sum _{i\ne r} f_i\lambda _i^2 +\left( \sup _{{\bar{M}}}|{\underline{\lambda }}|+\frac{1}{\epsilon }\right) \sum _{i=1}^n f_i +\sum _{i=1}^n f_{i}({\underline{\lambda }}_i-\lambda _i), \text{ if } \lambda _r\le 0. \end{aligned} \end{aligned}$$(45)As a result, we can remove assumption (28) in Theorem 7 and then improve existence results.
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Acknowledgements
The author wants to thank Professor Bo Guan, Professor Chunhui Qiu and Professor Xi Zhang for their constant encouragement. The author is supported by the National Natural Science Foundation of China (Grant No. 11801587).
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Communicated by F.H. Lin.
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