Blow Up Analysis For Solutions Of The Liouville Type Equations With Exponential Neumann Boundary Condition

The Liouville type equation has been widely used in mathematical physics problems and geometric analysis,for instance,the problem of mean field equation,prescribing Gaussian curvature and Chern-Simons theory,etc.In the past few decades,mathematic researchers have made many important achievements in the research of Liouville type equation,which greatly promoted the progress of mathematics and physics.For the research of the Liouville type equation,an important aspect is blow-up analysis,since it is related to the the convergence of sequence of solutions.By the trick of blow-up analysis,we can treat Moser-Trudinger equality,the local uniform estimate and problems of existence of solution,etc.For the Liouville type equation with Dirichlet boundary condition,many researchers studied this type equation and made a lot of achievements.But in recent years,arising in prescribing Gaussian curvature and boundary curvature problem,the Liouville type equation under exponential Neumann boundary condition has also attracted the attention of mathematic researchers and have some progresses for this type equation without singular data.But when interior or the boundary has singularity,the research of Liouville type equation with exponential Neumann boundary condition and with singular data has important significance.On the base,we mainly analyze this type equation.The specific content of this article can be summarized as follows.In the first chapter,the research background of this paper and the research progress of the Liouville type equation with exponential Neumann boundary condition are briefly introduced,and the main contents of this paper are also introducedIn the second chapter,we introduce the quantization property of the blow up value for the Liouville type equation with exponential Neumann boundary condition without singular data.We will use the "sup+inf" inequality to establish Li-Shafrir's type energy quantization resultIn the third chapter,we introduce the blow up analysis for Liouville type equation with exponential Neumann boundary condition and with singular data And we give the quantization property of the blow up value for the singular blow up point.When the equation has singular data,the study of the equations will be more difficult.We give the version of Tarantello's decomposition Lemma and new "sup+inf" type inequality under exponential Neumann boundary conditions separately.In the fourth chapter,We establish uniform estimates for blow-up sequence of solutions near an isolated singular blow up point.We treat two cases ?(?)N and??N+ separately,and obtain two different uniform estimates.