Inhomogeneous Schrödinger system with inverse square potential in three space dimensions

• Published in: Journal of inequalities and applications Vol. 2025; no. 1; pp. 97 - 29
• Main Authors: Boulaaras, Salah, Ghanmi, Radhia, Saanouni, Tarek
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2025; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Blow-up; Energy; Estimates; Gagliardo-Nirenberg inequality; Global existence; Ground state; Inequality; Inhomogeneous Schrödinger coupled system; Inverse square potential; Investigations; Mathematics; Mathematics and Statistics; Nonlinear equations; Norms; Nonlinear equations; Inverse square potential; 35Q55; Global existence; Gagliardo-Nirenberg inequality; Inhomogeneous Schrödinger coupled system; Blow-up
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-025-03352-0

This work studies the inhomogeneous Schrödinger coupled system with inverse square potential i ∂ t u j + Δ u j − λ | x | 2 u j = ± | x | − τ ( ∑ k = 1 m a j k | u k | p ) | u j | p − 2 u j . In this paper, u j : R × R 3 → C and the above parameters satisfy 1 ≤ j ≤ m , τ > 0 , 1 ≤ p < 3 − τ and λ > − 1 4 . In order to avoid the singular term | u j | p − 2 , one assumes that p ≥ 2 . The invariant Sobolev norm under the classical scaling ∥ κ 2 − τ 2 ( p − 1 ) u j ( κ 2 t , κ ⋅ ) ∥ H ˙ s c = ∥ u j ( κ 2 t ) ∥ H ˙ s c gives the energy critical exponent 3 − τ , which corresponds to 1 = s c : = 3 2 − 2 − τ 2 ( p − 1 ) . So, necessarily, one needs to assume that 0 < τ < 1 . The assumption λ > − 1 4 is motivated by the critical Hardy inequality 1 4 ∫ R 3 | f ( x ) | 2 | x | 2 d x ≤ ∫ R 3 | ∇ f ( x ) | 2 d x . In the stationary regime, one proves a Gagliardo-Nirenberg estimate adapted to the above problem and one establishes the existence of ground states. In the evolution regime, one develops an energy local theory. Moreover, for small datum, a local solution extends to a global one which scatter in the energy space. Eventually, in the inter-critical regime, one obtains a dichotomy of global existence versus blow-up of energy solutions under the ground state threshold. The limit case τ → 0 gives some results about the local/global well-posedness of the homogeneous Schrödinger system with inverse square potential in the energy space. Indeed, to the authors knowledge, such a problem was not treated before.