Normalized solutions to the Schrödinger systems with double critical growth and weakly attractive potentials

• Published in: Electronic journal of qualitative theory of differential equations Vol. 2023; no. 42; pp. 1 - 22
• Main Authors: Long, Lei, Feng, Xiaojing
• Format: Journal Article
• Language: English
• Published: University of Szeged 01.01.2023
• Subjects: normalized solutions; positive solutions; schrödinger systems; weakly attractive potentials
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2023.1.42

In this paper, we look for solutions to the following critical Schrödinger system { − Δ u + ( V 1 + λ 1 ) u = | u | 2 ∗ − 2 u + | u | p 1 − 2 u + β r 1 | u | r 1 − 2 u | v | r 2 i n   R N , − Δ v + ( V 2 + λ 2 ) v = | v | 2 ∗ − 2 v + | v | p 2 − 2 v + β r 2 | u | r 1 | v | r 2 − 2 v i n   R N , having prescribed mass ∫ R N u 2 = a 1 > 0 and ∫ R N v 2 = a 2 > 0 , where λ 1 , λ 2 ∈ R will arise as Lagrange multipliers, N ⩾ 3 , 2 ∗ = 2 N / ( N − 2 ) is the Sobolev critical exponent, r 1 , r 2 > 1 , p 1 , p 2 , r 1 + r 2 ∈ ( 2 + 4 / N , 2 ∗ ) and β > 0 is a coupling constant. Under suitable conditions on the potentials V 1 and V 2 , β ∗ > 0 exists such that the above Schrödinger system admits a positive radial normalized solution when β ⩾ β ∗ . The proof is based on comparison argument and minmax method.