来自Toda 系统, 二阶和四阶共形协变椭圆偏微分方程中有很多重要且具有挑战性的课题, 它们在物理与几何等领域中有着深厚的背景和应用. 这些方程和方程组吸引了很多学者的兴趣, 数学家们对其做出了大量杰出工作.这些方程(组) 的共同点之一是它们的解均表现一定的能量集中爆破现象, 研究爆破解对于理解它们至关重要. 本文旨在运用爆破分析和椭圆方程经典理论等理论方法来研究这些方程(组) 爆破解的性质, 并给出部分应用.首先, 对于紧黎曼曲面上奇异二阶平均场方程, 我们考虑了当爆破点全部为奇异点或者爆破点同时包含奇异点和正则点两种情形下爆破解的唯一性问题. 我们证明了若奇异点处的Dirac 测度强度不是4π 的整数倍, 爆破解是唯一的. 这是对Lin-Yan 在文献[Adv. Math. 338 (2018) 1141–1188] 中和Bartolucci 等在文献[J. Math. Pures Appl. (9) 123 (2019) 78–126] 中结果的推广.其次, 对于紧黎曼流形上正则?𝑈(3) Toda 系统, 我们首次研究了当参数?𝑘1 ,?𝑘2 均趋于临界值时方程组爆破解的性质; 具体来讲, 参数趋于临界值是指(?𝑘1, ?𝑘2 ) →(4π, 4?π) 或者(𝜌?1 , 𝜌?2 ) → (4?π, 4π), 其中? 是正整数(双临界情形). 我们证明了爆破行为只有三种可能, 并对每一种爆破行为得到了?𝑘1 ? 4π 和?𝑘2 ? 4𝑁π 的主项, 爆破点的位置估计, 以及不同爆破区域爆破高度的精细比较. 此外, 基于上述分类和估计, 本文证明了在适当的假设条件下爆破解的行为只能是第一种情形. 这是首次关于双临界?𝑈(3) Toda 系统爆破解的研究. 这些结果将是未来计算解的拓扑度以及研究无穷远处临界点理论的重要前提和工具.最后, 我们考虑了4 维紧黎曼流形上奇异预定?-曲率方程, 我们通过在奇异点附近做爆破分析研究爆破解序列的能量集中现象, 得到了解序列的渐近性; 该结果表明爆破解序列在奇异爆破点附近以点测度的形式收敛, 与此同时我们得到了该点测度的强度量化值, 也即能量积分的极限值.

There are quite a few important and challenging projects in Toda system, secondorder and fourth order conformally covariant partial differential equations. They have profound background and great applications in the field of physics and geometry. Many famous mathematicians have been drown into the research of these equation(s) and madeplanty of remarkable work. The solutions to all these equation(s) may exhibit certain concentration and blow-up phenomena, which means that it is crutial to analyse their bubbling solutions. In this thesis, we aim to study the blow-up properties of these bubbling solutions via the blow-upanalysis and elliptic equations theories, and give some applications.First of all, for singular mean field equations defined on a compact Riemann surface, we consider the uniqueness property of bubbling solutions in two cases: (1) all blow-up points are singular sources, and (2) some blow-up points coincide with the singularities of the Dirac data. If the strength of the Dirac mass at each singular blow-up point is not a multiple of 4π, we prove that bubbling solutions are unique. This work extends previous results of Lin-Yan [Adv. Math. 338 (2018) 1141– 1188] [1] and Bartolucci, et, al [J. Math. Pures Appl. (9) 123 (2019) 78–126] [2].In addition, for regular ?𝑈(3) Toda systems defined on Riemann surface, we initiate the study of bubbling solutions if parameters (?𝑘1, ?𝑘2) are both tending to critical positions: (?𝑘1, ?𝑘2 ) → (4π, 4?π) or (4?π, 4π) (? > 0 is an integer). We prove that there are at most three formations of bubbling profiles, and for each formation we identify leading terms of ?𝑘1 ? 4π and ?𝑘2 ? 4𝑁π, locations of blow-up points and comparison of bubbling heights with sharp precision. In addition, we prove that the first case is the only possibility of formations of bubbling solutions under certain assumptions. The results of this article will be used as substantial tools for a number of degree counting theorems, critical point at infinity theory in the future.In the end, we consider the prescribing ?−curvature equation defined on compact4?manifold (?, 𝑔). We study the concentration phenomenon by means of blow-up analysis. Consequently, we obtain that the sequence of bubbling solutions converges to point measure and calculate the exact value of the energy, which is also the strength of such point measure.