Let $u: \mathbb{R}^2 \to \mathbb{R}.$ Suppose I have a solution to the equation


$$\Delta(u)+e^{-u} \geq 0$$ on $\mathbb{R}^2$. Let r be the radial coordinate on $\mathbb{R}^2$. Suppose that $$lim\_{r \to \infty} u \to c$$ 


where $c$ is some constant. Can I get some estimate on the maximum of $u$ over all of $\mathbb{R}^2$ ? I'm sorry if my question admits an easy solution, but I am not usually working on analysis and couldn't find this precise situation in the literature. 



"C^0 estimate for solutions to $\Delta(u)+e^{-u} \geq 0$"
**One cannot bound $\sup u$ in terms of $c$.** To see this, set $u(x) = c + \alpha \,\varphi(\varepsilon x)$, where $\varphi\in S(\mathbb R^2;\mathbb R)$, $\varphi(0)=1$, and $\varphi\leq1$, while $\alpha>0$, $\varepsilon>0$ are parameters. Notice that
$$
 \Delta u(x) + e^{-u(x)} \geq e^{-\alpha} \left( e^\alpha\alpha\varepsilon^2 
 \inf\Delta\varphi + e^{-c} \right).
$$


Now, given $K>c\_+$, let $\alpha=K-c>0$ and choose $\varepsilon>0$ in
such a way that $e^\alpha\alpha\varepsilon^2 \inf\Delta\varphi +
e^{-c}\geq0$. Then $\sup u= u(0)=K$ and $\Delta u + e^{-u}\geq0$;
so we have the desired example.


