Existence And Asymptotic Behavior Of Solutions For Hénon Type Systems

In this paper, we study the existence and asymptotic behavior of ground state solutions for Henon type systems. This paper is constituted with four chapters.In chapter 1, we introduce the background and main results about Henon type systems. We illustrate difficulties about the existence and asymptotic behavior of solutions for Henon type systems, ideas to solve them, and main results.In chapter 2, we consider the existence of nontrivial solutions about the following Henon type systems: where B1(0) is a unit ball in RN, p, q> 1, p+q< 2*:=2N/N-2,2* is the critical Sobolev exponent. By the best imbedding constant achieved, we prove that problem (2) exists nontrivial solutions. Moreover, problem (2) exists at least two nontrivial solutions when a is large enough (one is radial, another is nonradial).In chapter 3, we consider the limiting behavior of the ground state solution (u, v) as p+qâ†'2*. We prove that the maximum points of two components u and v concentrate at the same point on the boundary (?) B1(0). Moreover, the ground state solution of problem (2) is non-radial as p+q near 2*.In chapter 4, we consider the limiting behavior of the ground state solution (u, v) as aâ†'+∞. We prove that the maximum points of two components u and v concentrate at the same point on the boundary (?) B1(0). Moreover, the ground state solution of problem (2) is non-radial as a large enough.Above results have been formed two papers:1.Existence and asymptotic behavior of solutions for Henon type systems has been submitted.2.Asymptotic behavior of ground state solution for Henon type systems have been published in Electron. J. Diff. Equ., Vol.2010(2010), No.116, pp.1-14.