Blow-up Solutions for Mean Field Equations and Toda Systems on Riemann Surfaces

Abstract

This dissertation studies mean field type equations and the SU(3) Toda system, significant topics both in mathematical physics and differential geometry, with wide-ranging applications. Its primary focus lies in the construction of mixed boundary–interior bubbling solutions on Riemann surfaces with boundary, which exhibit blow-up at prescribed numbers of points in the interior and on the boundary as the parameters approach critical values. For mean field type equations and partial blow-up solutions of Toda systems, variational methods and Lyapunov-Schmidt reduction are employed to construct the solutions. However, for asymmetric blow-up solutions of Toda systems, due to the intricate limit profiles, the Lyapunov-Schmidt reduction cannot be applied directly. Herein, we introduce the “k-symmetric” condition for the surface and utilize singular perturbation methods to construct a family of bubbling solutions that blows up at “k-symmetric centers” of the surface. The dissertation is organized into two parts: one exploring blow-up solutions for mean field type equations and the other for the SU(3) Toda system. Each part delves into the construction of blow-up solutions under various scenarios.