Subsolution theorem for the Monge-Ampère equation over almost Hermitian manifold

Let $\Omega\subseteq M$ be a bounded domain with smooth boundary $\partial\Omega$, where $(M,J,g)$ is a compact almost Hermitian manifold. Our main result of this paper is to consider the Dirichlet problem for complex Monge-Ampère equation on $\Omega$. Under the existence of a $C^{2}$-smooth strictly $J$-plurisubharmonic ($J$-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolution which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds. %This work was already done by Pliś when we assume there is a strictly $J$-psh defining function for $\Omega$.