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Given condition: Let $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$. Suppose there exist $w(>0)\in H_{0}^1(\Omega)$ satisfying the inequality $-\Delta w\leq C e^w$ in $\Omega$ for some positive constant $C$. Can you please help me with the following question.

Does there exist some function $v\in H_{0}^1(\Omega)\cap C^1(\Omega)$ such that $w\leq v$ and satisfying $-\Delta v=Ce^{v}$ in $\Omega$.

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