一类拟线性椭圆方程组正弱解的存在性和不存在性

摘要

本文研究了一类拟线性椭圆方程组在有界区域上的边值问题的正弱解的存在性和不存在性。利用方程组的特征值和特征函数分别构造了弱上解和弱下解，通过比较原理证明了弱下解小于弱上解，用上下解方法得到了正弱解的存在性。最后，用反证法证明了正弱解的不存在性，并给出了正弱解的不存在性条件。%In this paper, the existence and nonexistence of positive weak solutions for the quasilinear elliptic systems with boundary value problem in bounded domain are investigated. Firstly, we establish the weak upper and lower solutions of equations by the eigenvalue and eigenfunction, and prove that the weak lower is less than upper solutions by the comparison principle. Then according to the method of weak upper and lower solutions, we obtain the existence of positive weak solution for the systems. Finally, we derive the nonexistence of positive weak solutions by proof of contradiction, and give the nonexistence condition of positive weak solutions.