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Asymptotic radial symmetry for solutions of \(\Delta u+e^ u=0\) in a punctured disc. (English) Zbl 0794.35049
Summary: A representation formula for solutions of the equation \[ \Delta u+2Ke^ u=0,\quad K\text{ a constant}, \tag{*} \] in a punctured disc in terms of multi-valued meromorphic functions is found. As application it is deduced that a necessary and sufficient condition for a solution of \((*)\), \(K>0\), being asymptotic radially symmetric, is \(\int e^ u<\infty\).

MSC:
35J60 Nonlinear elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
30D30 Meromorphic functions of one complex variable (general theory)
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