Energy-critical scattering for focusing inhomogeneous coupled Schrodinger systems

This work investigates the time asymptotics of the inhomogeneous coupled Schrodinger equations i partial derivative(t)u(j) + Delta u(j) = vertical bar x vertical bar(-rho) (Sigma(1 <= k <= m) a(jk)vertical bar u(k)vertical bar(p)) vertical bar u(j)vertical bar(p-2)u(j), where j is an element of [1, m], rho > 0, and u(j) : R-t x R-x(3) -> C. Here, one treats the energy-critical regime u(0, center dot) is an element of [H-1(R-N)](m) and 1 = s(c) := N/2 - 2-rho/2(p-1). This is the index of the invariant Sobolev norm under the dilatation parallel to lambda 2-rho/2(p-1) u(lambda(2)t, lambda center dot)parallel to((H) over dotsc) lambda(mu-N/2+2-p/2(p-1))parallel to U(lambda(2)t)parallel to((H) over dotsc). To the authors knowledge, the technique used in order to prove the scattering of an energy global solution to the above problem in the sub-critical regime S < S-c is no more applicable for S = S-c. In order to overcome this difficulty, one uses the Kenig-Merle road map. In order to avoid a singularity of the source term, one considers the case p >= 2, which restricts the space dimensions to N = 3. Moreover, in order to use the Sobolev injection (H) over dot(1) hooked right arrow L2N/N-2, one restricts the space dimensions to N = 3. Compared with the previous work for the first author (Inhomogeneous coupled non-linear Schrodinger systems. J. Math. Phys. 62, 101508 (2021)), the method consisting on dividing the integrals in the unit ball and its complementary seems not sufficient to conclude in the present study because of the energy-critical exponent. Instead, one uses some Caffarelli-Kohn-Nirenberg weighted type inequalities.