Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent

Abstract

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.