2007 Fiscal Year Final Research Report Summary
Research on the structure of solutions for anisotropic quasilinear elliptic equations
| Project/Area Number |
17540197
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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 | Naruto University of Education |
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Principal Investigator |
NARUKAWA Kimiaki Naruto University of Education, College of Education, Professor (60116639)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUOKA Takashi NARUTO UNIVERSITY OF EDUCATION, College of Education, Professor (50127297)
TORISU Ichiro NARUTO UNIVERSITY OF EDUCATION, College of Education, Associate Professor (50323134)
ITO Masayuki Tokushima University, Faculty of Integrated Arts and Sciences, Professor (70136034)
FUKAGAI Nobuyoshi Tokushima University, Faculty of Engineering, Associate Professor (90175563)
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| Project Period (FY) |
2005 – 2007
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| Keywords | anisotropic differential equation / Orlicz-Sobolev space / quasilinear elliptic equation / variational inequatlity / positive solution / p(x)-Lanlacian / concentration-compactness / biburcation theory |
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| Research Abstract |
1. We have considered quasilinear elliptic equations which have princial parts with ordinary growth rate and exterior forces of critical growth in the sense of the Orlicz-Sobolev inequality. Several results on the existence of nonnegative and nontrivial solutions for this type of equations are obtained. Further, under the assumption of uniform ellipticity besides, the regularity of solutions have been proved. By using the regularity, it is shown that the strong maximum principle is valid for nonnegative solutions. This fact implies the existence of positive solutions. Furthermore the structure of global bifurcation of positive solutions has been obtained.
2. In the case when principal parts grow very slowly the Orlicz-Sobolev space naturally given by the attached functional to the differential equation is not refrexive, and further the energy functional is not Frechet differentiable. This causes the difficulty of analysis for this type of equations. We have investigated these equations with exterior forces with critical growth and given the existence of nonnegative, nontrivial solutions. In the proof, we have used the mountain pass lemma for variational inequality and concentration-compactness argument by P. L. Lions.
3. When p for p-Laplacian depends on the space variable x, that is p(x)-Laplacian, it has been investigated compared with p-Lapalacian. If the rate of variation is sufficiently small, then nontrivial solution behaves similarly to the one of p-Laplacian. However it is expected that the structure is different completely in general. We have not yet obtained sufficient results.
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Research Products
(8 results)
