M.Sc Student | Lorena Aguirre |
---|---|

Subject | On Semi-Linear Elliptic Systems Exhibiting Critical Behavior |

Department | Department of Applied Mathematics |

Supervisor | Full Professor Shafrir Itai |

For N≥3 and *m *a positive
integer, let A=[a_{ij}] 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

-∆u_{i} =(∑ _{jϵI
}a_{ij} u_{j} c_{i })^{p*}

∫ (∑ _{jϵI }a_{ij}
u_{j} c_{i })^{ p*}=M_{i}

^{ }

with boundary conditions

u_{i}=0 on
∂Ω (if Ω is bounded)

for i=1,?,*m*,
where Ω is either a bounded smooth domain or the whole space R^{N},
the vector of finite positive masses M=(M_{i})ϵ(R_{})^{m
}is given, and c=(c_{i})ϵR^{m }is an unprescribed
vector of real constants. When is a bounded domain, we are looking for
(u,c)ϵ(H^{2}(Ω)∩H_{0}^{1}(Ω))^{m}?R^{m}
satisfying (S) almost everywhere (we later show that such function u is a
classical solution); if =R^{N}, the notion of solution is taken in the
classical sense, i.e., uϵ(C^{2}(R^{N}))^{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 R^{N}. 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 R^{N}, 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 R^{N}. This question is stated in the Introduction,
after the discussion of the last Theorem we obtained.