Let $(\Sigma, g_0)$ be a closed, oriented surface with a Riemannian metric satisfying $\int_\Sigma dA_0 = 1$, where $dA_0$ is the area form on $\Sigma$, and let $\nabla_0$ and $K_0$ denote the gradient and the Gaussian curvature of $(\Sigma, g_0)$, respectively. Consider all the metrics conformal to $g_0$ that preserve the area, i.e., all metrics of the form \begin{equation*} \{e^{2\varphi} g_0: \varphi \in C^\infty(M)\} \end{equation*} such that \begin{equation*} \int_\Sigma e^{2\varphi} \, dA_0 = 1, \end{equation*} and define the following functional $F(\varphi)$ on the set of all such area-preserving conformal metrics: \begin{equation*} F(\varphi) = \frac{1}{2}\int_\Sigma |\nabla_0 \varphi|^2 \, dA_0 + \int_\Sigma K_0 \varphi \, dA_0 - \pi \chi(\Sigma) \log\left(\int_\Sigma e^{2\varphi} dA_0\right). \end{equation*} My question is: For the case $\chi(\Sigma) \leq 0$, how do we show that $F$ is bounded from below?
For context, this functional is from the following paper that involves extremizing a functional of the metric on a Riemannian manifold and, in particular, I'm trying to understand why the infimum of $F$ exists: https://www.sciencedirect.com/science/article/pii/0022123688900705