2001 Fiscal Year Final Research Report Summary
Studies on Some Degenerate Quasilinear Elliptic Equations in Unbounded Domains
| Project/Area Number |
11640207
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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, 学校教育学部, 教授 (60116639)
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| Co-Investigator(Kenkyū-buntansha) |
ITO Masayuki Tokushima University, Faculty of integrated Arts and Sciences, Professor, 総合科学部, 教授 (70136034)
MATSUOKA Takashi Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (50127297)
MURATA Hiroshi Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (20033897)
FUKAGAI Nobuyoshi Tokushima University, Faculty of Engineering, Associate Professor, 学校教育学部, 教授 (90175563)
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| Project Period (FY) |
1999 – 2001
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| Keywords | quasilinear elliptic equation / bifurcation theory / p-Laplacian / nonlinear eigenvalue problem / multiple positive solution / minimal solution / unbounded domain / mini-max principle |
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| Research Abstract |
We have investigated some properties of nontrivial solutions of quasilinear degenerate elliptic equations which are equal to p-Laplacians asymptotically at the origin and infinity respectively. The results obtained in this project are as follows:
First we have considered the case when the asymptotic orders of the principal part and the term of the exterior force are equal at the origin and the infinity respectively. By giving the the uniform estimate of the gradients of solutions for the equations and by generalizing the degree theory of Leray-Schauder and global bifurcation theory by Rabinowitz, we have obtained the structure of the branches of positive solutions bifurcating from the trivial solution and infinity. Namely, positive solutions bifurcate from the zero solution and the infinity at the first eigenvalues of the limitting p-Laplacians at zero and infinity respectively. Further bifuracation phenomina from the higher eigenvalues are also given. Differed from the linear equations
… More
, it is not trivial to obtain higher eigenvalues for the p-Laplacian. The Ljusternik-Schnirelman theory is used to construct a series of eigenvalues. Although it is not known that the system of eigenfunctions corresponding to those eigenvalues are complete or not, we have shown that nontrivial solutions bifurcate at least at those eigenvalues from the trivial solution and infinity.
Secondary the case when the orders of the principal part and the term of the exterior force are different has been investigated. By using the first bifurcation results stated above, we have given the the multiplicity and nonexistence of positive solutions for each parameters and existence of the minimal solutions. In the argument the structure of the branch of positive solutions for the equations discussed at first plays a crucial role. We showed the comparison theorem and extend the principle of "H^1 versus C^1 local minimizers" given by Brezis-Nirenberg to degenerate qusilinear elliptic equations of this type. Applying these results, we obtain some properties of positive solutions stated above. Less
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