2006 Fiscal Year Final Research Report Summary
Quasilinear Elliptic Differential Equations of Critical Nonlinear Growth
| Project/Area Number |
16540197
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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 | The University of Tokushima |
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Principal Investigator |
FUKAGAI Nobuyoshi The University of Tokushima, Institute of Technology and Science, Associate Professor, ソシオテクノサイエンス研究部, 助教授 (90175563)
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| Co-Investigator(Kenkyū-buntansha) |
NARUKAWA Kimiaki Naruto University of Education, Department of Mathematics, Professor, 学校教育学部, 教授 (60116639)
ITO Masayuki The University of Tokushima, Faculty of integrated Arts and Sciences, Professor, 総合科学部, 教授 (70136034)
KOHDA Atsuhito The University of Tokushima, Institute of Technology and Science, Associate Professor, ソシオテクノサイエンス研究部, 助教授 (50116810)
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| Project Period (FY) |
2004 – 2006
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| Keywords | quasilinear / elliptic equation / variational method / positive solution / Sobolev's exponent / Orlicz space / Orlicz-Sobolev space / concentration-compactness |
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| Research Abstract |
We studied a Dirichlet boundary value problem of a degenerate quasilinear elliptic equation which has the φ-Laplace operator in the principal part. The main result is the existence theorem of nonnegative nontrivial solutions via variational methods in Orlicz-Sobolev space settings. It can be applied to a wide class of elliptic equations even if the principal parts have non power-like nonlinearities. In the following let φ(t)t = Φ'(t)
(1) The quasilinear elliptic problem of subcritical growth: An existence theorem of multiple nonnegative nontrivial solutions is proved. In a previous paper we have discussed the problem under the hypothesis φ(t)t = o(f(x, t)) at t = 0 and ∞, which corresponds to classical results about a semilinear elliptic equation with a concave-convex lower term. In this time we consider the case f(x, t) = ο(φ(t)t) at t = 0 and ∞ contrary to the problem treated above. Further we also consider an equation with more general principal part.
(2) The quasilinear elliptic problem of critical Orlicz-Sobolev growth : We make some modification of the standard concentration-compactness principle and obtain an existence theorem of a nonnegative nontrivial solution. For example it can be applied to the the case Φ(t) = t^p log(1 + t), p > 1.
(3) Minimax problem of a nonsmooth functional : The variational problem for a functional with slowly growing principal part and involving critical Orlicz-Sobolev lower term (with respect to the principal part) is discussed. The functional is not Frechet differentiable, although it Gateaux differentiable. A nonnegative nontrivial solution for the Euler equation is given. For example the result can be applied to the case Φ(t) = t log(1 + t).
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Research Products
(6 results)
