一类具有退化性和奇异性的拟线性椭圆方程正解的注记

Some Notes on the Positive Solutions for a Class of Quasi-linear Elliptic Equations with Degeneracy and Singularity

讨论一类主部为p-Laplace的具有退化性和奇异性的拟线性椭圆方程的正解.给出了在空间维数为一时所论问题有无界解的条件,以及在高维空间时,区域充分规则和星形情形下,所论边值问题在Sobolev空间无解的条件.运用函数变换等技巧,克服了由于退化性和奇异性带来的困难,对p≠2时的某些结论做了进一步补充.

Positive solutions of quasi-linear elliptic equations with degeneracy and singularity for a class of p-Laplace problems are discussed. For one dimension, the conditions of boundless solutions are given. And for multi-dimension the conditions of the problems with sufficiently regular and star-shaped domain are also given, which have no solutions in Sobolev space. The techniques including functional transformation were used to overcome the difficulties coming from the degeneracy and singularity of the problems. A further complementarity of the equations for p ≠ 2 is given, too.