M.Sc Thesis
M.Sc Student Aguirre Lorena On Semi-Linear Elliptic Systems Exhibiting Critical Behavior Department of Applied Mathematics Professor Itai Shafrir

Abstract

For N≥3 and m a positive integer, let A=[aij] be an m?m matrix with non-negative entries, and define p*:=N/(N-2). We are interested in studying the following semi-linear elliptic system of equations

-∆ui =(∑ jϵI aij uj c)p*

∫ (∑ jϵI aij uj c) p*=Mi

with boundary conditions

ui=0      on ∂Ω      (if Ω is bounded)

for i=1,?,m, where Ω is either a bounded smooth domain or the whole space RN, the vector of finite positive masses M=(Mi)ϵ(R)m is given, and c=(ci)ϵRm is an unprescribed vector of real constants. When is a bounded domain, we are looking for (u,c)ϵ(H2(Ω)∩H01(Ω))m?Rm satisfying (S) almost everywhere (we later show that such function u is a classical solution); if =RN, the notion of solution is taken in the classical sense, i.e., uϵ(C2(RN))m.

Our work consists of four main parts. In the first part, we prove some general results about solutions to the problem in a ball and in the space RN. Among these results, a Liouville-type Theorem is remarkable. Then, we find conditions on the matrix A that ensure that for any M and c, each of the components of a function u satisfying (S) in RN, is radially symmetric and decreasing about some point. Thirdly, existence of solutions to the problem in bounded domains is established by using a dual formulation of the problem, through minimization. Lastly, we show it is possible to derive existence of solutions to the problem in bounded domains from existence of solutions to the problem in the whole space, and in the other direction, we can assure existence of entire solutions for the limiting mass for which the functional involved in the dual formulation is bounded from below, by employing a blow-up argument.

We are left with a question at the moment we do not know how to solve. It concerns a monotonicity property for the set of masses for which there exists a solution to the problem in RN. This question is stated in the Introduction, after the discussion of the last Theorem we obtained.