具分数阶Laplace算子的非线性椭圆方程解的存在性

Existence of solutions for nonlinear elliptic equation with the fractional Laplacian

摘要

摘要:为了研究一类具分数阶Laplace算子的非线性椭圆方程解的存在性问题,首先利用积分等价方程和压缩映射原理证明了该方程解的存在唯一性,进一步地,通过取特殊值和恰当的分段函数再利用Matlab编程得到了该方程在(0,0,0)处数值解的近似值. 最后利用Picard序列、数学归纳法和反证法证明了方程的解的一些性质.

Abstract:In order to study the existence of solutions for a class of nonlinear elliptic equation with the fractional Laplacian. Firstly, the existence of a unique solution for this equation can be proved by using the integral equivalent equation and the contraction mapping principle. Secondly, the approximate value of its numerical solution at (0,0,0) are obtained by taking special values, appropriate piecewise function and using matlab. Finally, using Picard sequence, induction and the method of contradiction to prove the properties of the solution.