Abstract.
In this paper, we study the asymptotic behavior of solutions of the Dirichlet problem for the Liouville equation ¶¶\( -\Delta u= \lambda {{K(x)e^u} \over {\int_{\Omega}K(x)e^u}} \)¶¶on a bounded smooth domain \( \Omega \) in the plane as \( \lambda \to 8m\pi \), where \( m=1,2,... \). The equation is also called the Mean Field Equation in Statistical Mechanics. By a result of H. Brezis and F. Merle, any solution sequence may have a finite number of bubbles. We give a necessary condition for the location of the bubble points.
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Received: December 3, 1999; revised version: September 10, 2000
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Ma, L., Wei, J. Convergence for a Liouville equation. Comment. Math. Helv. 76, 506–514 (2001). https://doi.org/10.1007/PL00013216
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DOI: https://doi.org/10.1007/PL00013216