Solution structures of nonlinear elliptic PDEs and new perspective of qualitative theory

2014 Fiscal Year Final Research Report

• Project/Area Number: 24740100
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Global analysis
• Research Institution: The University of Tokyo (2013-2014); Keio University (2012)
• Principal Investigator: MIYAMOTO Yasuhito 東京大学, 数理(科)学研究科(研究院), 准教授 (90374743)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 優臨界 / 分岐理論 / 非線形楕円型方程式 / 非線形解析 / ノイマン問題 / ディリクレ問題 / Joseph-Lundgren指数

Outline of Final Research Achievements

I studied the structure of the positive solutions of nonlinear elliptic PDEs. In particular, I studied bifurcation diagrams. Mainly, two results were obtained. I proved that the imperfect bifurcation occurs in the bifurcation diagram of the positive solutions of the Gel'fand problem when the domain is perturbed. I almost completely classified the bifurcation diagrams of the positive radial solutions of the Dirichlet problem of supercritical elliptic PDEs in a ball. I obtained partial results about the bifurcation diagram of the positive radial solutions of the Neumann problem of the supercritical scalar field equation in a ball.

Free Research Field

大域解析学