Monotonicity Of Solutions For System Involving Fractional P-Laplacian

Fractional p-Laplace equation is a nonlinear partial differential equation is of great significance.With the development of science and technology,the fractional partial differential equation by the attention of many researchers,it can completely describe the field of science of genetic effects,and many phenomena of the memory.In recent years,the fractional partial differential equation has been widely applied.In this paper,we study the monotonicity of solutions for system involving fractional p-Laplacian Here Ω is a domain(bounded or unbounded)in Rn which is convex in x1 direction.with nonlinear terms f(x,u(x),▽u(x)),based on the sliding method,We show that the solutions ui(x)are monotone increasing with respect to x1 in Ω under various conditions on the nonlinear terms.The appearance of ▽ui(x)in the nonlinear terms brings in some new difficulties,the more subtle method is needed to deal with this problem.In addition,we introduce a new iteration method to deal with the sysetm which can be applied to study many other problems.This paper consists of three chapters:In chapter one,we give the background,related research,main work and preliminary knowledge of fractional p-Laplacian equations.In chapter two,we using sliding method and narrow field theorem to study the monotony of the solution of fractionalp-Laplace equations with ▽u(x)in the bounded region Ω about x1 direction convex.In chapter three,we using sliding method,narrow field theorem and iterative method to study the monotony of the solution of fractional-p-Laplace equations with ▽u(x)in the unbounded band Ω with respect to x1 direction.