# Existence of solution to Laplace equation

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 $$\Omega$$.