Regularity, Symmetry And Monotonicity Of Positive Solutions For Two Types Of Nonlinear Partial Differential Systems

In this paper, we study the regularity, symmetry and monotonicity of positive solutions of thefollowing nonlinear partial diferential systemsThis paper is constituted with four chapters. In chapter1, we introduce the background andmain results about problems (1)and (2). We illustrate difculties in the study of the regularity,symmetry and monotonicity of positive solutions and ideas to solve them. We also contain mainresults.In chapter2, we recall the basic properties of the kernels of operators (+id)α2and()α2+id. The symmetry, monotonicity and decay of these two kernels make the symmetry, monotonicityand regularity of the positive solutions of fractional partial diferential systems under these twofractional partial operators possible.In chapter3, we first introduce in the weighted Hardy-Littlewood-Sobolev Inequality related toGαwhich is the kernel of operator (+id)α2, this inequality could replace the comparison theoremand make the properties of solutions provable. By the equivalence between the positive solutionsof fractional partial diferential systems under Bessel operator and the positive solutions of integralsystems under the Bessel potential, we then prove the regularity, symmetry and monotonicity ofpositive solutions of the integral systems under the Bessel kernel, which could lead to the regularity,symmetry and monotonicity of positive solutions of the fractional partial diferential systems underBessel potential operator.In chapter4, we first introduce in the weighted Hardy-Littlewood-Sobolev Inequality relatedto the Kαwhich is the kernel of operator ()α2+id, then we use the same methods as chapter3to study the main properties of positive solutions of the fractional partial diferential systemsunder Bessel-alike potential operator.