Existence of solutions for nonlinear elliptic equation with the fractional Laplacian

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.