  We analyze the existence of K\"ahler-Einstein metrics of positive curvature
in the neighborhood of a germ of a log terminal singularity $(X,p)$. This boils
down to solve a Dirichlet problem for certain complex Monge-Amp\`ere equations.
We show that the solvability of the latter is independent of the shape of the
domain and of the boundary data. We establish a Moser-Trudinger $(MT)_{\gamma}$
inequality in subcritical regimes $\gamma<\gamma_p$ and establish the existence
of smooth solutions in that cases. We show that the expected critical exponent
$\hat{\gamma}_p=\frac{n+1}{n} \widehat{\mathrm{vol}}(X,p)^{1/n}$ can be
expressed in terms of the normalized volume, an important algebraic invariant
of the singularity.
