Normalized solutions of coupled Sobolev critical Schrodinger equations with mass subcritical couplings

2025/07/17 by Jianjun, Zhang, Xuexiu Zhong, Xuexiu, Zhong +2
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Advanced Numerical Methods in Computational Mathematics #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2507.13163

Abstract

We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schrödinger equations \begincases -Δu+λ₁ u=μ₁|u|^{2^*-2}u+να|u|^α-2|v|^βu ~\hboxin~ \R^N,
-Δv+λ₂ v=μ₂|v|^{2^*-2}v+νβ|u|^α|v|^β-2v ~\hboxin~ \R^N \endcases subject to the mass constraints ∫_ℝ^N|u|² \ud x=a² and ∫_ℝ^N|v|² \ud x=b², where, a>0, b>0, N=3,4 and 2^*:=(2N)/(N-2) is the Sobolev critical exponent. The main purpose of this paper is focused on the mass mixed case, i. e., α>1,β>1,α+β<2+(4)/(N). For some suitable small ν>0, we show that the above system admits two positive solutions, one of which is a local minimizer, and another one is a mountain pass solution. Moreover, as ν→0⁺, asymptotic behaviors of solutions are also considered. Our result gives an affirmative answer to a Soave's type open problem raised by Bartsch \it et al. (Calc. Var. Partial Differential Equations 62(1), Paper No. 9, 34, 2023).

Citations

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• Positive normalized solutions of Schrödinger equations with Sobolev critical growth in bounded domains
• Normalized solutions for a class of Sobolev critical Schrodinger systems