Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent - Caju, Rayssa Helena Aires de Lima (2018)
In this work we study the asymptotic behavior to positive solutions of the following coupled elliptic system of nonlinear Schrödinger equations ∆gui − 2 X j=1 Aij(x)uj + n(n−2) 4 |U| 4 n−2ui = 0 which are deﬁned in the punctured unit ball B1(0)\{0} for n ≥ 3. Here g is a Riemannian metric on the unit ball and the potential A is assumed a C1 map such that Aij(x) is a symmetrical matrix for each x in B1(0). From the viewpoint of conformal geometry, this systems are pure extensions of Yamabe-type equations. We will approach the problem assuming ﬁrst that g is the euclidian metric and the potential A vanishes. In this case we are able to prove that the solutions of our problem are asymptotics to what we call Fowler-type solutions. In the general case we will prove the same result by putting some restrictions on the potential and assuming that the dimension is less or equal to ﬁve.