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Quasilinear Elliptic Differential Equations of Critical Nonlinear Growth

临界非线性增长的拟线性椭圆微分方程

基本信息

批准号：
16540197
负责人：
FUKAGAI Nobuyoshi
金额：
$ 2.37万
依托单位：
The University of Tokushima
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

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).

研究了一类主算子为φ-拉普拉斯算子的退化拟线性椭圆型方程的Dirichlet边值问题.主要结果是在Orlicz-Sobolev空间中利用变分方法得到了非负非平凡解的存在性定理。它可以应用到一个广泛的一类椭圆方程，即使主部分有非幂类非线性。设φ（t）t = φ ′（t）（1）次临界增长的拟线性椭圆型问题：证明了多个非负非平凡解的存在性定理。在前一篇文章中，我们讨论了假设φ（t）t = o（f（x，t））在t = 0和∞时的问题，这对应于半线性椭圆型方程的经典结果。此时我们考虑f（x，t）= o（φ（t）t）在t = 0和∞时的情况，与上面处理的问题相反。进一步，我们还考虑了一个主部更一般的方程. (2)临界Orlicz-Sobolev增长的拟线性椭圆问题：对标准的集中紧性原理作了一些修改，得到了非负非平凡解的存在性定理.例如，它可以应用于Φ（t）= t^p log（1 + t），p > 1的情况。(3)非光滑泛函的极大极小问题：讨论了主部缓慢增长且包含临界Orlicz-Sobolev低阶项（关于主部）的泛函的变分问题。泛函不是Frechet可微的，尽管它是Gateaux可微的。给出了欧拉方程的一个非负非平凡解。例如，结果可以应用于Φ（t）= t log（1 + t）的情况。