Research on canonical forms to nonlinear elliptic boundary value problems and the global structure of all solutions including singular solutions

首先，首席调查员S。Yotsutani得到了半线性椭圆型问题径向解的标准形和结构定理。Kabeya和E.半线性椭圆问题的径向解满足二阶微分方程的某些边值问题。结果表明，通过适当的变量变换，边值问题可化为Dirichlet、Neumann或Robin边界条件下的标准型。得到了具有幂非线性项和各种边界条件的方程的标准型的结构定理。利用这些定理，可以系统地研究半线性椭圆型方程径向解的性质，明确各种方程的未知结构;其次，可以通过标准形将一个问题的结果转化为另一个问题的结果，并找到一种研究奇异解的新方法。作为一个具体的例子，S. Yotsutani阐明了H的单位球中的解的结构，包括奇异解。Myogahara和E.作为相关问题，我们正在研究数学生物学中出现的交叉扩散方程的极限方程。我们证明了它有不同类型的奇异解。卢和W。M.倪该问题是一个非局部非线性椭圆边值问题，目前还没有一种求解方法，我们发现了一种新的求解方法。有很多有趣的问题，该方法是适用的。奥森螺旋流问题就是其中之一。

First, a head investigator S. Yotsutani have obtained canonical forms and structure theorems for radial solutions to semilinear elliptic problems with Y. Kabeya and E. Yanagida.Radial solutions of semilinear elliptic problems satisfy some boundary value problems for second order differential equations. It is seen that the boundary value problems can be reduced to a canonical form with the Dirichlet, Neumann or Robin boundary condition after suitable change of variables. We get structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.Second, it is possible through the canonical forms to convert results for one problem to that of others, and moreover, to find an original methods to investigate singular solutions. As a concrete example, S. Yotsutani clarified the structure of solutions including singular solutions in a unit ball with H. Myogahara and E. Yanagida.As related problems, we are investigating a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions with Y. Lou and W.-M. Ni. This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method. There are a lot of interesting problems for which the method is applicable. A problem of the Ossen's spiral flow is one of them.