Studies on Some Degenerate Quasilinear Elliptic Equations in Unbounded Domains
Project/Area Number 
11640207

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  Naruto University of Education 
Principal Investigator 
NARUKAWA Kimiaki Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (60116639)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Period (FY) 
1999 – 2001

Project Status 
Completed (Fiscal Year 2001)

Budget Amount *help 
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2000: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1999: ¥800,000 (Direct Cost: ¥800,000)

Keywords  quasilinear elliptic equation / bifurcation theory / pLaplacian / nonlinear eigenvalue problem / multiple positive solution / minimal solution / unbounded domain / minimax principle / 正値解 / 多重解 
Research Abstract 
We have investigated some properties of nontrivial solutions of quasilinear degenerate elliptic equations which are equal to pLaplacians 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 LeraySchauder 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 pLaplacians 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 pLaplacian. The LjusternikSchnirelman 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 BrezisNirenberg 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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Research Products
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