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 |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year 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)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
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 + u|u|^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 non-radically symmetric solution exists we also showed how many non-radically 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)
