2018-01-012024-05-14https://scholars.lib.ntu.edu.tw/handle/123456789/661074摘要：This proposal aims to study a class of nonlinear elliptic semilinear equation (or system). This class of PDE often possesses the socalled “bubbling phenomenon”. The problems to characterize when the bubbling phenomenon occure and to determine the behaviors near any bubbling points are extremely difficult problems. One of features is the limiting equation (or systems of equation), after suitably scaling, tends to an elliptic PDE, which also possesses some specific “integrability”. This means that a local solution (often with some singularities) of this limiting equation can be expressed by meromorphic datas, which mechanism of the expression might be very complicated. So we call any equation of this class of nonlinear semi-linear equation to be integrable elliptic PDEs. Examples are the well-known curvature equation (or Liouville`s equation) and Toda system associated with a simple Lie algebra. This class of equations have bean arised from geometry and physics naturally. Our proposal is pioneering in studying these equations by combining ideas and methods from different research areas in mathematics. <br> Abstract: This proposal aims to study a class of nonlinear elliptic semilinear equation (or system). This class of PDE often possesses the socalled “bubbling phenomenon”. The problems to characterize when the bubbling phenomenon occure and to determine the behaviors near any bubbling points are extremely difficult problems. One of features is the limiting equation (or systems of equation), after suitably scaling, tends to an elliptic PDE, which also possesses some specific “integrability”. This means that a local solution (often with some singularities) of this limiting equation can be expressed by meromorphic datas, which mechanism of the expression might be very complicated. So we call any equation of this class of nonlinear semi-linear equation to be integrable elliptic PDEs. Examples are the well-known curvature equation (or Liouville`s equation) and Toda system associated with a simple Lie algebra. This class of equations have bean arised from geometry and physics naturally. Our proposal is pioneering in studying these equations by combining ideas and methods from different research areas in mathematics.可積分系統kdv方程非線性偏微分方程integrable systemsKdV equationscurvature equation