Studies on the Structure of Solutions of Degenerate Quasilinear Elliptic Equations
Project/Area Number  09640197 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Naruto University of Education 
Principal Investigator 
NARUKAWA Kimiaki Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (60116639)

CoInvestigator(Kenkyūbuntansha) 
FUKAGAI Nobuyoshi Tokushima University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90175563)
MURATA Hiroshi Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (20033897)
MATSUNAGA Hiromichi Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (30032634)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,900,000 (Direct Cost : ¥1,900,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1997 : ¥1,000,000 (Direct Cost : ¥1,000,000)

Keywords  quasilinear elliptic equation / pLaplacian / bifurcation theory / limit eigenvalue problem / viscosity solution / infinity Laplacian / 準線形楕円型方程式 / pラプラシアン / 分岐理論 / 極限固有値問題 / 粘性解 / 無限大ラプラシアン / 準線形退化型楕円型方程式 
Research Abstract 
We have considered quasilinear degenerate elliptic equations which are equal to pLaplacians asymptotically at tile origin and infinity. Some results concerning to the aspects of solutions for the equations of this type are obtained as are listed below. 1. First we have obtained the structure of the branches of positive solutions bifurcated from the trivial solution and tile infinity solution. Namely, positive solutions bifurcate from the zero solution and the infinity solution at the first eigenvalues of the pLaplacians which the considered equation are equal to aymptotically at zero and infinity respectively. Furthermore, in the case when the growth orders of the principal part of the equation at zero and infinity, these branches are identical. 2. Next the asymptotic behavior of eigenvalues and eigenfunctions of pLapalace operators as the power p tends to infinity has been investigated. We have obtained the best constant of Poincare's inequality at p=*, and further, given a limit eigenvalue problem which characterize tile limits of these eigenvalues and eigenfunctions. Althogh the equation ill the limit eigenvalue problem is a fully nonlinear elliptic equation, a pair of the limits of eigenvalues and eigenfunctions solves this limit eigenvalue problem in the sense of viscosity. 3. Finally we have considered the equation defined in the whole space instead of a bounded domain. In this case, tile similar results as are stated in I.are also valid under some assumptions on the potential part of the nonlinear operator. It is our problem to loose these restrictions hereafter.

Report
(4results)
Research Output
(9results)