2006 Fiscal Year Final Research Report Summary
Research of the structure of unbounded viscosity solutions to semilinear degenerate elliptic equations in R^N
Project/Area Number 
16540151

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Kobe University 
Principal Investigator 
MARUO Kenji Kobe University, Faculty of Maritime Sciences, Professor, 海事科学部, 教授 (90028225)

CoInvestigator(Kenkyūbuntansha) 
ISHII Katuyuki Kobe University, Faculty of Maritime Sciences, Associate Professor, 海事科学部, 教授 (40232227)
KAGEYAMA Yasuo Kobe University, Faculty of Maritime Sciences, Lecturer, 海事科学部, 講師 (70304136)

Project Period (FY) 
2004 – 2006

Keywords  Semilinear Elliptic Differential Equations / Structure of Solutions / Viscosity Solutions 
Research Abstract 
Consider a following semilinear degenerate partial differential equation: g(x)Δu + uu^p  f(x) = 0 ∈ R^N (1) where g is a nonnegative plynominal of degree ell > 2 in a neighborhood of a point at infinity and f is also a plynominal in a neighborhood of a point at infinity. Moreover, assume g holds bounded zero points. Hence, the differential equation is a degenerate tyle. We don't impose the boundary condition to solutions of (1) in the neighborhood of the point infinity. Then, It is possible to exist many continuous viscosity solutions. Our purpose of this research was to analyze the structure of the set of many continuous viscosity solutions of (1). The results of our research are as follows. We the first showed that an inequality of relations of N and k is a necessary and sufficient condition to decide whether radically symmetric solutions are infinite or single where k is a coefficient of a maxima order of f. Moreover, we proved that a set of many radically symmetric solutions is homeomorphic to R^1. The secondly, under the assumptions N = 2 and that lower order terms of polynominal of f, g do not exist, we found the condition to judge whether there exists non radically symmetric solution or not. If nonradically symmetric solution exists we also showed how many nonradically symmetric solutions there were. That is, We showed that the number of solutions was different depending on the value that related to l, p, k.

Research Products
(8 results)