一个具退化强制的Laplace方程有界弱解存在性

Existence of bounded weak solutions to a Laplace equation with degenerate coercivity

借助方程低阶项的正则化效应,得到了解的最大模估计.运用偏微分方程中的弱收敛方法,证明了椭圆方程有界弱解的存在性.应用此类方程解的结果和证明方法,可以进一步研究具一阶梯度项的椭圆方程、拟线性的具低阶项的p-Laplace方程以及带有零阶项的抛物方程等弱解的存在性,也可以进一步研究方程解的局部有界性.

With the help of the regularizing effect of the lower order term,the maximal norm estimate to the solutions is obtained.Based upon the weak convergence methods for partial differential equations,the existence of bounded weak solutions to an elliptic equation is testified.Employing the results and the methods used in this class of equations,the further investigation for the bounded weak solutions to other type of equations can be done.For instance,the elliptic equations with the first order gradient term,the quasi-linear p-Laplace equations with lower order term,the parabolic equations with zero order term and so on.Moreover,it is applied to the study of the local boundedness of the solutions.