Research On The Properties Of Solutions Of Semilinear Elliptic Problems With Nonlocal Operators And Degenerate Operators

The research in this paper is divided into two parts.In the first part,we mainly use the classical moving plane method to study the radial symmetry and monotone of the positive solutions in the unit sphere and the whole space by constructing the maximum principle,the narrow region principle and the decay at infinity principle of the weighted fractional Laplace operator,In addition,since the dual operator of the fractional Hardy operator is the weighted fractional Laplace operator under special circumstance,we study the properties of the positive solution of fractional Hardy problem by using the symmetry results of the positive solution of the semilinear problem of the weighted fractional Laplace operator.In the second part,we mainly study the weighted L∞ estimates for Poisson problems involving Hardy-Leray operator-Δ+μ/|x|2 or elliptic operators with critical singular gradients-Δu+τ+/|x|2x·▽.Our method overcomes the difficulty from the singular potentials.Moreover,this estimates could be applied to obtain global W1,p estimates for distributional solution of Hardy problem with the Radon source.Next,we introduce the structure of this paper.In chapter 1,we introduce the background of the problem and main results of the thesis.In chapter 2,we analyze the symmetry and monotone of the positive solutions for semilinear problems involving the weighted fractional Laplacians.In chapter 3,we show the global estimates for Poisson problems with singular potential.