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.
