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In this paper we prove some Calabi-Bernstein type and non-existence resultsconcerning complete $[\varphi,\vec{e}_{3}]$-minimal surfaces in$\mathbb{R}^{3}$ whose Gauss maps lie on compacts subsets of open hemispheresof $\mathbb{S}^{2}$. We also give a general non-existence result for completespacelike $[\varphi,\vec{e}_{3}]$-maximal surfaces in $\mathbb{L}^{3}$ and, inparticular, we obtain a Calabi-Bernstein type result when $\dot{\varphi}$ isbounded.

Calabi-Bernstein type results for critical points of a weighted area functional in $\mathbb{R}^{3}$ and $\mathbb{L}^{3}$