具有非线性奇异项和变指数的拟线性椭圆问题解的存在性

摘要

In this paper,we proved the existence of the solutions for the Dirichlet boundary value problem of quasilin-ear elliptic equation with singular term and variable exponent. Firstly, we constructed an approximation problem, using Sobolev embedding theorem and the supremum and infimum of the variable exponent to overcome difficulties arising from singular term, thus we prove the boundedness of the solution sequence for the approximation problem, then we solved the difficuties caused by p-Laplace operator by selecting the suitable test functions and a priori estimate tech-nique, and with the help of the boundedness of solution sequence for the approximation problem, the sufficient condi-tions of the existence of solutions for this problem are obtained. By contrast,the approximation method we used in this paper is better than the upper and lower solution method in the past.%针对于具有奇异项和变指数的拟线性椭圆方程Dirichlet边值问题,给出了证明该问题解的存在性的方法.首先构造一个逼近问题,利用Sobolev嵌入定理和变指数的上下确界,克服了来自奇异项和变指数的困难,证明了逼近问题解序列的有界性,然后通过选取适当的检验函数和先验估计技巧克服了来自p-Laplace算子的困难,再借助于逼近问题解序列的有界性,得到了该问题解存在的充分条件.通过对比,采用的逼近方法要优于以往常用的上下解方法.