Asymptotic Expansions of Solutions of the Yamabe Equation near Isolated Singular Points

We study asymptotic behaviors of positive solutions to the Yamabe equation near isolated singularities and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, Gidas, and Spruck, and a work by Korevaar, Mazzeo, Pacard, and Schoen. Then, we study the existence of solutions with prescribed asymptotic expansions near singular points and an arbitrarily high order of approximation.