Quasilinear Elliptic Differential Equations of Critical Nonlinear Growth

研究了主体为φ-Laplace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。它可以应用于一类广泛的椭圆型方程，即使其主项具有非幂类非线性。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。在以前的文章中，我们讨论了假设φ(T)t=o(f(x，t))在t=0和∞下的问题，这对应于关于具有凹凸下项的半线性椭圆型方程的经典结果。此时，我们考虑的情况是f(x，t)=ο(φ(T)t=t=0，∞与上面讨论的问题相反。此外，我们还考虑了一个具有更一般主部的方程。(2)临界Orlicz-Sobolev增长的拟线性椭圆型问题：我们对标准的浓度-紧性原理作了一些修正，得到了一个非负非平凡解的存在定理。例如，它可以应用于Φ(T)=t^plog(1 T)，p&gt；1的情形。(3)非光滑泛函的极大极小问题：讨论了具有缓慢增长的主项且涉及临界Orlicz-Sobolv下项(相对于主项)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，该结果可以应用于案例Φ(T)=t log(1 T)。

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).