# Regularity of a nonlinear PDE

1. May 29, 2012

### Goklayeh

Could someone give me some hint (or some reference) about the study of regularity of weak solutions of
$$(*)\quad \begin{cases} -\Delta u = e^u & \text{in }\Omega\\ u = 0 & \text{on }\partial \Omega \end{cases}$$
where $\Omega \subset \mathbb{R}^2$ is a bounded domain with smooth boundary (here (as usual), a weak solution of (*) is a function $u \in H^1_0(\Omega)$ which satisfies $\int_{\Omega}{\nabla u \cdot \nabla \varphi\mathrm{d}x} = \int_{\Omega}{e^u \varphi \mathrm{d}x}\:\: \forall \varphi \in H^1_0(\Omega)$).