Critical inhomogeneous coupled Schrödinger equations

Abstract

This work develops a local theory of the inhomogeneous coupled Schrödinger equations iu̇j+Δuj=σ|x|−γ∑1≤k≤majk|uk|p|uj|p−2uj,j∈[1,m]. Here, one treats the critical Sobolev regime u(0,⋅)∈[Hsc(RN)]m, where sc≔N2−2−γ2(p−1) is the index of the invariant Sobolev norm under the dilatation ‖λ2−γ2(p−1)u(λ2t,λ⋅)‖Ḣsc=λμ−N2+2−γ2(p−1)‖u(λ2t)‖Ḣsc. 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 < sc, which consists of dividing the integrals on the unit ball of RN 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 |x|−γ∈LNγ,∞, 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 |x|−γ. In order to avoid a singularity of the source term, one considers the case p ≥ 2, which restricts the space dimensions to N ≤ 3.