2006 Fiscal Year Final Research Report Summary
Study on Positive Solutions of Nonlinear Elliptic Boundary Value Problems Arising in Population Dynamics
| Project/Area Number |
16540165
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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 |
Basic analysis
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| Research Institution | Maebashi Institute of Technology |
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Principal Investigator |
UMEZU Kenichiro Maebashi Institute of Technology, Faculty of Engineering, Associate Professor, 工学部, 助教授 (00295453)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Nonlinear elliptic problem / Nonlinear boundary condition / Multiplicity / Bifurcation equation / Principal eigenvalue / Variational technique / Population dynamics |
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| Research Abstract |
In this research we studied existence and behavior of positive solutions to a nonlinear elliptic boundary value problem of logistic type in a smooth bounded domain of the Euclidean space, as a parameter included varies.
1. We established a theorem on bifurcation and stability of positive solutions under nonlinear boundary conditions. A local bifurcation analysis was carried out by using the Lyapunov-Schmidt type of procedure, the local bifurcation theory due to Crandall and Rabinowitz and the method of super and subsolutions. Moreover we reduced our problem to a constrained minimization one based on the first bifurcation solution, to obtain multi plicity of positive solutions.
2. We established a nonexistence theorem for positive solutions under nonlinear boundary conditions when parameter is small enough, where a variational technique and super and subsolutions are used.
3. We studied a linear indefinite eigenvalue problem with a Robin type boundary condition. Existence and uniqueness of a positive principal eigenvalue was proved by use of a variational approach. We also obtained an a priori lower bound for the positive principal eigenvalue in terms of the sign-changing weight included in the given eigenvalue problem. We remark that we can get the linear eigenvalue problem under consideration if the nonlinear elliptic boundary value problem of logistic type is linearized at the trivial solution which is identically equal to zero.
4. We studied blowing-up property of the positive principal eigenvalue for a linear elliptic eigenvalue problem with an indefinite weight and Neumann boundary condition. Necessary and sufficient conditions for the blowing-up property were discussed. It should be emphasized that we constructed a counterexample to show that a known necessary and sufficient condition for the blowing-up property in the Dirichlet boundary condition case no longer remains true.
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