Suppose $u$ is a smooth solution of the following PDE on $S^2$: $$ -\frac{1}{3}\Delta_{S^2} u+1=e^{2u}. $$ Prove that the PDE only admits a constant solution, i.e., $u\equiv 0$ on $S^2$.
With loss of generality, we can assume $u=u(x_3)$, and by the classic stereographic projection, we can transform the PDE to an ODE in $\mathbb{R}^2$. Thus to prove the result, we may rely on some energy estimate for a function related to $u$. I believe a candidate function could be $G(x_3)=(1-x_3)\partial_{x_3}u$. But I am stucked in the process.
Could you give some suggestions? Appreciate any help!