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

Long, Lei and Feng, Xiaojing (2023) Normalized solutions to the Schrödinger systems with double critical growth and weakly attractive potentials. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (42). pp. 1-22. ISSN 1417-3875

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Abstract

In this paper, we look for solutions to the following critical Schrödinger system −∆u+(V1+λ1)u = |u|2∗−2u+|u|p1−2u+βr1|u|r1−2u|v|r2 in RN, −∆v+(V2+λ2)v = |v|2∗−2v+|v|p2−2v+ βr2|u|r1|v|r2−2v in RN, having prescribed mass RN u2 = a1 > 0 and RN v2 = a2 > 0, where λ1,λ2 ∈ R will arise as Lagrange multipliers, N 3, 2∗ = 2N/(N −2) is the Sobolev critical exponent, r1,r2 > 1, p1, p2,r1 + r2 ∈ (2 +4/N,2∗) and β > 0 is a coupling constant. Under suitable conditions on the potentials V1 and V2, 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.

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