Qualitative Theory of Nonlinear Elliptic Differential Equations
Project/Area Number  09640196 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  University of Tokushima 
Principal Investigator 
FUKAGAI Nobuyoshi The University of Tokushima, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90175563)

CoInvestigator(Kenkyūbuntansha) 
KOHDA Atsuhito University of Tokushima, Faculty of Engineering, Associate Professor, 工学部, 助教授 (50116810)
NAITO Manabu University of Ehime, Faculty of Science, Professor, 理学部, 教授 (00106791)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  nonlinear / elliptic / differential equations / qualitative theory / quasilinear / weak solutions / bifurcation / eigenvalue problems / 非線形 / 楕円型 / 微分方程式 / 定性理論 / 準線形 / 弱解 / 分岐 / 固有値問題 / 楕円形 
Research Abstract 
We studied the subjects related to qualitative theory of nonlinear elliptic differential equations : (i) boundary value problems of quasilinear elliptic equations in a bounded domain ; (ii) oscillatory problem of ordinary differential equations which derived from qualitative problem of elliptic equations in an unbounded domain. Our results are the following. [1] The asymptotic behavior of eigenvalues and eigenfunctions of pLaplace operator is investigated. We obtain the best constant of L*Poincare inequality, and a limit equation which the limits of eigenvalues and eigenfuncitons satisfy in a weak sense. [2] A SturmLiouville equation on (a, *) is examined. Supposing a strongly nonoscillatory condition, we obtain a sequence of positive princial eigenvalues and the corresponding principal eigenfunctions. [3] Concering an initial value problem of parabolic equation, we obtain a new sufficient condition on the initial value which determines the solution to blow up. Furthermore, we investigate the asymtotic behavior of the solution at the blowing lip time. [4] We consider a second order halflinear differential equation on [a, *]. We take up the two classes of nonoscillatory solutions (i.e., princilal solutions and non principal solutions), and show that precise information can be drawn as to how the number of zeros of these solutions changes as lambda varies from zero to infinity. [5] A bifurcation problem or a nonlinear eigenvalue problem for degenerate quasilinear elliptic equations with Dirichelt boundary condition is studied. By virture of our estimates, we can apply the LerayShauder degree theory to our problem and obtain the bifurcation of nontrivial weak solutions. [6] Quasiperiodic solutions to Van der Pol type equations driven by two or more distinct freequency input signals are considered. The existence and uniqueness results are obtained from the viewpoint of numerical analysys.

Report
(4results)
Research Output
(22results)