Symmetry And Regularity Of Positive Solutions To Integral Systems

In this thesis, we investigate the symmetry and regularity of positive solutions of two kinds of integral systems with Bessel potential. This thesis is divided into three chapters.In chapter 1, we introduce the background of the problem and main results.Chapter 2 is concerned with the symmetry and regularity of positive solutions of the following integral system with Bessel potential, where Gα(x) is the a order Bessel kernel, n≥3,0< β<α<n,1<p, q, and We show that every positive solution triple (u, v, w) is radially symmetric and monotonic decreasing about some point by the moving planes method in integral forms. Moreover, by the regularity lifting method, we prove that(u,v,w) belongs to L∞(Rn) × L∞(Rn) × L∞(Rn), and which is then locally Holder continuous.Chapter 3 is concerned with the symmetry and regularity of positive solutions of the following double weighted integral system where 0≤β, τ,β+τ<α<n,1<p,q<n-β/β and Similarily, we show that every positive solution triple (u, v) is radially symmetric and monotonic decreasing about some point by the moving planes method in integral forms.