Energy‐critical scattering for focusing inhomogeneous coupled Schrödinger systems.

This work investigates the time asymptotics of the inhomogeneous coupled Schrödinger equations i∂tuj+Δuj=|x|−ρ∑1≤k≤majk|uk|p|uj|p−2uj$$ \mathrm{i}{\partial}_t{u}_j&#x0002B;\Delta {u}_j&#x0003D;{\left&#x0007C;x\right&#x0007C;}&#x0005E;{-\rho}\left(\sum \limits_{1\le k\le m}{a}_{jk}{\left&#x0007C;{u}_k\right&#x0007C;}&#x0005E;p\right){\left&#x0007C;{u}_j\right&#x0007C;}&#x0005E;{p-2}{u}_j $$, where j∈[1,m],ρ>0$$ j\in \left[1,m\right],\rho >0 $$, and uj:ℝt×ℝx3→ℂ$$ {u}_j:{\mathrm{\mathbb{R}}}_t\times {\mathrm{\mathbb{R}}}_x&#x0005E;3\to \mathrm{\mathbb{C}} $$. Here, one treats the energy‐critical regime u(0,·)∈[H1(ℝN)]m$$ u\left(0,\cdotp \right)\in {\left[{H}&#x0005E;1\left({\mathrm{\mathbb{R}}}&#x0005E;N\right)\right]}&#x0005E;m $$ and 1=sc:=N2−2−ρ2(p−1)$$ 1&#x0003D;{s}_c:&#x0003D; \frac{N}{2}-\frac{2-\rho }{2\left(p-1\right)} $$. This is the index of the invariant Sobolev norm under the dilatation ‖λ2−ρ2(p−1)u(λ2t,λ·)‖H˙sc=λμ−N2+2−ρ2(p−1)‖u(λ2t)‖H˙sc$$ {\left\Vert {\lambda}&#x0005E;{\frac{2-\rho }{2\left(p-1\right)}u}\left({\lambda}&#x0005E;2t,\lambda \cdotp \right)\right\Vert}_{{\dot{H}}&#x0005E;{s_c}}&#x0003D;{\lambda}&#x0005E;{\mu -\frac{N}{2}&#x0002B;\frac{2-\rho }{2\left(p-1\right)}}{\left\Vert u\left({\lambda}&#x0005E;2t\right)\right\Vert}_{{\dot{H}}&#x0005E;{s_c}} $$. 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<sc$$ s<{s}_c $$ is no more applicable for s=sc$$ s&#x0003D;{s}_c $$. 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$$ p\ge 2 $$, which restricts the space dimensions to N≤3$$ N\le 3 $$. Moreover, in order to use the Sobolev injection H˙1↪L2NN−2$$ {\dot{H}}&#x0005E;1\hookrightarrow {L}&#x0005E;{\frac{2N}{N-2}} $$, one restricts the space dimensions to N=3$$ N&#x0003D;3 $$. Compared with the previous work for the first author (Inhomogeneous coupled non‐linear Schrödinger 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. [ABSTRACT FROM AUTHOR]