Critical exponents line, scattering thoeries for a weighted gradient system of semilinear Schrodinger equations

Abstract

In this paper, we consider the following Cauchy problem of a weighted(or essential) gradient system of semilinear Schrödinger equationsUnder certain assumptions, we establish the local wellposedness of the H 1 × H 1 -solution, Ḣ1 × Ḣ1 -solution and H s × H s -solution of the system with different types of initial data.If f (|u| 2 , |v| 2 )u = λ|u| α |v| β+2 u and g(|u| 2 , |v| 2 )v = µ|u| α+2 |v| β v with λ > 0, µ > 0, α ≥ 0 and β ≥ 0, it is surprised that there exists a critical exponents line α + β = 2 when d = 3 in the following sense: The system always has a unique bounded H 1 × H 1 -solution for any initial data (u0, v0) ∈ H 1 (R 3 ) × H 1 (R 3 ) if α + β ≤ 2, yet we can find some initial data (u0, v0) ∈ H 1 (R 3 ) × H 1 (R 3 ) such that it doesn't possess the global H 1 × H 1 -solution if α + β > 2 and α = β. While when d = 4, we call (α, β) = (0, 0) is the critical exponents point in the following sense: The system always has a unique bounded H 1 ×H 1 -solution for any initial data (u0, v0)Moreover, we establish the H 1 × H 1 and Σ × Σ scattering theories for the solution if α + β < 2(i.e., (α, β) is below the critical exponents line α + β = 2) when d = 3, Σ × Σ scattering theory for the solution if (α, β) is on the critical exponents line α + β = 2 excluding the endpoints when d = 3, Ḣ1 × Ḣ1 scattering theory for the solution if (α, β) is the endpoint of the critical exponents line α + β = 2 when d = 3 and (α, β) = (0, 0) when d = 4. We also establish Ḣsc × Ḣsc scattering theories for the corresponding solutions if (α, β) is above the critical exponents line α + β = 2 when d = 3 and (α, β) > (0, 0) when d ≥ 4. Here sc = d 2 − 2 α+β+2 .