中文摘要：

利用α-对称化方法证明了带混合边界条件的半线性椭圆方程的Raye-Rayner型不等式,并推出了相关线性椭圆方程混合边界条件正解的一些重要估计.

英文摘要：

In this paper, a Payne-Rayner type inequality for semilinear elliptic equations with mixed boundary condition is obtained by α-symmetrization method. Moreover, some estimates for the positive solution of a class of the related semilinear elliptic problem are given.