1999 Fiscal Year Final Research Report Summary
Studies on Eigenvalue Problems of Nonlinear Elliptic Equations
| Project/Area Number |
10640208
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Hiroshima University |
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Principal Investigator |
USAMI Hiroyuki Hiroshima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (90192509)
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| Co-Investigator(Kenkyū-buntansha) |
TANAKA Kazunaga Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (20188288)
YOSHIDA Kiyoshi Hiroshima University, Faculty of Integrated Arts and Sciences, Professor, 総合科学部, 教授 (80033893)
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| Project Period (FY) |
1998 – 1999
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| Keywords | eigenvalue problem / elliptic equation / positive solution |
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| Research Abstract |
(1) Eigenvalue Problems of Elliptic Equations : Two-parameter eigenvalue problems for semilinear elliptic equations are studied. We establish asymptotic properties of (variational) eigenvalues and eigenfunctions. Two-parameter Ambrosetti-Prodi problems are also studied. We investigate the relation between parameters and the number of solutions.
(2) Positive Solutions of Elliptic Equations : Semilinear second-order elliptic euations are considered in unbounded domains. We establish multiplicity results for positive solutions and uniqueness theorems for positive solutions.
(3) Positive Solutions of Quasilinear Ordinary Differential Equations : Quasilinear ordinary differential equations whose leading term is one-dimensionai pseudo-Laplacian are considered. We obtain asynrptotic representations of positive solutions. As an application of these results, we show existence of several types of positive solutions of exterior Dirichlet problems for quasilinear elliptic equations.
(4) Mathematical Models Describing Aggregation Phenomena of Molds : We consider self-similar solutions of parabolic systems introduced by Keller and Segel to describe aggregation phenomena of molds due to chemotaxis. We clarify the relation between parameters and the number of self-similar solutions.
(5) Nonnegative Nontrivial Solutions of Quasilinear Elliptic Equations and Elliptic Systems : We establish necessary and/or sufficient conditions for quasilinear elliptic equations, as well as quasilinear elliptic systems, to possess nontrivial nonnegative entire solutions. Several Liouville type theorems are also obtained.
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