Liouville Type Theorems For An Integral Equations System On A Half-space R_+~n

The Liouville-type theorem is an important problem when we study the integral equa-tion or the integral equations system, many experts and scholars pay close attention to it. In this paper, by using the method of moving planes in integral forms we obtain the Liouville-type theorem for an integral equations system on a half-space R+n. Next, we prove the equivalence between PDES and IES on the half space R+n.The thesis is organized as follows:In the first chapter, there is an overview of the background and development status of the integral equation and the integral equations system, in addition,simply introduced the work we do and the basic theory of knowledge involved in this paper.In the second chapter, we present some non-existence results for an integral system on a half-space R+n.There are two parts in this chapter. In the first part, we mainly study the Liouville-type theorem of an integral system (0-6) under local integrability conditions.Theorem2.1.3Suppose n/n-α<p≤n+α/n-α and n/n-α<q≤and n+α/n-α. If u∈Ln(p-1)/α(R+n) and v∈Llocn(q-1)/α(R+n) are a nonnegative solution of integral equation (0-6), then u=0. Here can be any real number between0and n if n>3. While for n=3, we require1<α<n. In the second part, we discuss the Liouville-type theorem as the exponent general defined.Theorem2.2.4Assume that1<p, q<∞,and there exist p1≥1and q1≥1such that Suppose that u∈LP1(Rn)∩L∞(Rn) and v∈Lq1(Rn+)∩L∞(Rn) is a pair of positive solutions of integral system(0-6), then both u and v are srictly monotone increasing with respect to variable xn.From the Theorem2.2.4, we obtain the nonexistence of positive solution pair for integral equations system (0-6) as follows:Theorem2.2.5Let (u, v) be a pair of positive solutions of (0-6) with1<p, q<∞,there exist p1≥1and q1≥1such that (0-7),(0-8),(0-9).Assume that u∈(Rn)∩L∞(Rn) and v∈Lq1(Rn+) n L∞(Rn) are nonnegtive,then u=v=0.In the third chapter, We derive that the equivalence between the integral equation system (0-6) and the PDEs by using of poly-harmonic properties of PDEs. At the last, the summary and the development prospects of equation were given.