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Regularity of a nonlinear PDE

  1. May 29, 2012 #1
    Could someone give me some hint (or some reference) about the study of regularity of weak solutions of
    [tex]
    (*)\quad
    \begin{cases}
    -\Delta u = e^u & \text{in }\Omega\\
    u = 0 & \text{on }\partial \Omega
    \end{cases}
    [/tex]
    where [itex]\Omega \subset \mathbb{R}^2[/itex] is a bounded domain with smooth boundary (here (as usual), a weak solution of (*) is a function [itex]u \in H^1_0(\Omega)[/itex] which satisfies [itex]\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)[/itex]).

    Thank you in advance for your time!
     
  2. jcsd
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