Suppose $\varphi$ is a radial (and radially decreasing) solution of
$$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$
with $ R>0$, and $\psi$ is a decreasing radial function satisfying $$ \int_{\partial B(o,r)}|\nabla \psi| ds \leq \int_{B(0,r)}e^{\psi}dx, \ \ \ \ \text{for all} \ \ r \in (0,R). $$
Suppose $\psi (R)=\varphi(R)$. Is it true that $\varphi (r)\geq \psi (r)$ for all $r \in (0,R)$, or $$ \int_{B(0,R)}e^{\varphi} dx \geq \int_{B(0,R)} e^{\psi} dx?$$
