Bubbling Solutions To Elliptic System And Higher Order Elliptic Equations

There are quite a few important and challenging projects in Toda system,second order 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 made plenty of remarkable work.The solutions to all these equation(s)may exhibit certain concentration and blowup 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-up analysis 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 4?,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 SU(3)Toda systems defined on Riemann surface,we initiate the study of bubbling solutions if parameters(?1k,?2k)are both tending to critical positions:(?1k,?2k)?(4?,4N?)or(4N?,4?)(N>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?1k-4? and ?2k-4N?.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 Q-curvature equation defined on compact 4-manifold(M,g).We study the concentration phenomenon by means of blow-up analysis.As a consequence,we obtain that the sequence of bubbling solutions converges to point measure and calculate the exact value of the energy.Later on,we prove the concentration-compactness theorem.Basing on such theorem,we gain the critical set for the global invariant when blow-up phenomenon occurs and then establish a priori estimate if such invariant is not in a critical position.