Singular limits in Liouville-type equations

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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 \(m \ge 1\)

Received: 11 February 2004, Accepted: 17 August 2004, Published online: 22 December 2004