| Project/Area Number |
11640216
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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 | Waseda University |
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Principal Investigator |
TANAKA Kazunaga Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (20188288)
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| Co-Investigator(Kenkyū-buntansha) |
SHIBATA Tetsutaro Hiroshima University, Faculty of Integrated Arts & Sciences, Associate Professor, 総合科学部, 助教授 (90216010)
KURATA Kazuhiro Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10186489)
OTANI Mitsuharu Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (30119656)
ADACHI Shinji Waseda University, School of Science and Engineering, Reseach Associate, 理工学部, 助手 (40339685)
NAKASHIMA Kimie Tokyo University of Fisheries, Associate Professor, 水産学部, 助教授 (10318800)
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| Project Period (FY) |
1999 – 2001
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| Keywords | variational problems / elliptic equatims / Hamiltonian systems |
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| Research Abstract |
We study the existence problems for nonlinear differential equations via variational methods. We mainly dealt with nonlinear elliptic problems and Hamiltonian systems.
1. We study the existence and multiplicity of positive solutions of nonlinear scalar field equations in unbounded domains. In particular, we are concerned with equations which depend on the space variable x and we investigate the effects of the inhomogeneity (dependence on the space variable x) on the set of solutions of the scalar field equation. We find a very delicate dependence ― very small inhomogeneity induces a big change in the set of soluitons ― and we find an example of nonlinear scalar field equation which has 4 positive solutions after very small but not zero pert perturbation.
2. We also consider the singular perturbation problems for nonlinear elliptic problems. We get 2 results : (a) for 1-dimensional setting we introduce a new finite dimensional reduction and we succeed to prove the existence of solutions w
… More
ith a cluster of interior or boundary layers for inhomogeneous Allen-Cahn type equations. We also succeed to prove the existence of solutions with a cluster of spikes for nonlinear Schrodinger equations. (b) We give a mountain pass characterization of positive solutions for a wide class of nonlinear elliptic equations. As an application, we show the existence of a spike solution for a wide class of nonlinear elliptic equations including asymptotically linear equations.
3. For Hamiltonian systems we deal with singular Hamiltonian systems with 2-body type singularities. In case the potential V(q) has more than 3 strong force type singularities we find a family of very complex solutions which are related with symbolic dynamical systems. We also deal with the case the set S of singularity is not a point and it has a positive volume. We consider the case where V(q) 〜 - 1/ dist (q, S)^α and we find the existence of non-collision solutions for all positive α > 0, that is, even for weak force case 0 < α < 2. Less
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