What is the regularity of weak solutions of the Liouville equation?

[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 $\int_{\Omega}{\nabla u \cdot \nabla \varphi\mathrm{d}x} = \int_{\Omega}{e^u \varphi \mathrm{d}x}\:\: \forall \varphi \in H^1_0(\Omega)$).

Thank you in advance for your time!

 

[gtoguy97]

This equation is known as the Liouville equation. Regularity of weak solutions has been studied extensively in the literature; see for example Theorem 1.2 in [1] and Theorem 5 in [2].[1] H. Brezis and L.A. Peletier, Asymptotics for elliptic equations involving exponential nonlinearities, Arch. Rational Mech. Anal. 71 (1979), pp. 143-184.[2] Y. C. Kwong and K. M. Mak, On the regularity of weak solutions of the Liouville equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), pp. 257-273.