Expansions for distributional solutions of the elliptic equation in two dimensions

Assume [Formula: see text] is a planar domain, and [Formula: see text] is a locally bounded distributional solution to the elliptic equation [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] and [Formula: see text] are real analytic functions defined on [Formula: see text] and the real line [Formula: see text], respectively. We establish asymptotic expansions of [Formula: see text] to arbitrary orders near [Formula: see text], which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the weighted Yamabe equation, and partly extends that of Guo–Wan–Yang on the Liouville equation in a punctured disc. Our method is a combination of a priori estimate and mathematical induction.