TITLE

Singular limits in Liouville-type equations

AUTHOR(S)
del Pino, Manuel; Kowalczyk, Michal; Musso, Monica
PUB. DATE
September 2005
SOURCE
Calculus of Variations & Partial Differential Equations;Sep2005, Vol. 24 Issue 1, p47
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We consider the boundary value problem $ \Delta u + \varepsilon ^{2} k{\left( x \right)}e^{u} = 0$ in a bounded, smooth domain $\Omega$ in $ \mathbb{R}^{{\text{2}}} $ with homogeneous Dirichlet boundary conditions. Here $$ \varepsilon > 0,k(x) $$ is a non-negative, not identically zero function. We find conditions under which there exists a solution $ u_{\varepsilon } $ which blows up at exactly m points as $ \varepsilon \to 0 $ and satisfies $ \varepsilon ^{2} {\int_\Omega {ke^{{u_{\varepsilon } }} \to 8m\pi } }% $ . In particular, we find that if $k\in C^2(\bar\Omega)$, $ \inf _{\Omega } k > 0 $ and $\Omega$ is not simply connected then such a solution exists for any given $m \ge 1$
ACCESSION #
17671685

 

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