Delta-extremal functions in $\mathbb{C}^n$

The article is devoted to properties of a weighted Green function. We study the $(\delta,\psi)$-extremal Green function $V^{*}_{\delta}(z,K,\psi)$ defined by the class $\mathcal{L}_{\delta}=\big\{u(z)\in psh(\mathbb C^{n}):\ u(z) \leqslant C_{u}+\delta\ln^{+}|z|, \ z\in\mathbb C^{n}\big\}, \ \delta>0.$ We see that the notion of regularity of points with respect to different numbers $\delta$ differ from each other. Nevertheless, we prove that if a compact set $K\subset\mathbb{C}^{n}$ is regular, then $\delta$-extremal function is continuous in the whole space $\mathbb C^{n}.$