Abstract
This work develops a local theory of the inhomogeneous coupled Schrodinger equations iu(j)+Delta u(1)=sigma vertical bar x vertical bar(-gamma)(Sigma(1 <= k <= m)a(jk)vertical bar u(k)vertical bar(p))vertical bar u(j)vertical bar p(-2)u(j),j is an element of[1,m]. Here, one treats the critical Sobolev regime u(0,.)is an element of[H-sc(R-N)](m), where s(c):=N/2-2-gamma/2(p-1) is the index of the invariant Sobolev norm under the dilatation parallel to gamma(2-gamma/2(p-1))u(gamma(2)t,lambda.)parallel to(sc)(H(over dot)) = lambda mu(-N/2+2-gamma 2/(p-1)parallel to)u(lambda(2)t)||(.)(H)sc. To the authors knowledge, the technique used in order to prove the existence of an energy local solution to the above-mentioned problem in the sub-critical regime s = s(c), which consists of dividing the integrals on the unit ball of R-N and its complementary, is no more applicable for s = sc. In order to overcome this difficulty, one uses two different methods. The first one consists of using Lorentz spaces with the fact that vertical bar x vertical bar(-gamma) is an element of (L) over bar (N/gamma,infinity), which allows us to handle the inhomogeneous term. In the second method, one uses some weighted Lebesgue spaces, which seem to be suitable to deal with the inhomogeneous term vertical bar x vertical bar(-gamma). In order to avoid a singularity of the source term, one considers the case p >= 2, which restricts the space dimensions to N <= 3.