一类(p,q)-Laplacian椭圆方程组解的存在性

The existence of positive weak solution for a class of (p,q)-Laplacian elliptic system

研究了p,q-Laplacian椭圆方程组-△pu=λf(x)usvn,-△qv=μg(x)utvm的Dirichlet边界值问题的正弱解的存在性.首先根据两个方程组构造了弱上解和弱下解,然后利用弱上下解方法得到了方程组正弱解的存在性.利用特征值和特征函数构造了弱上下解具有一定的创新性,结果推广了p=q=2的情况,且对于任意的参数λ,μ>0方程组一定存在正解,并非需要充分大的参数.

In this paper, we discuss the existence of positive weak solution for the （p, q）- Laplacian and elliptic systems -△pu=λf（x）u^8v^n,-△qv=μg（x）u^tv^mwith Dirichlet boundary condition. First, we construct a weak lower solution and super solution according to two elliptic systems. Next, by using method of weak upper and lower solutions, we get the existence of positive weak solution for the systems. It is innovative to construct the upper and lower solutions using the eigenvalue and eigenfunction. Our result extends the case of p = q =- 2 . Further, the systems have a positive weak solution for any λ,μ〉0, not for sufficiently large λ,μ〉0.