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Spiked vector solutions of coupled Schrödinger systems with critical exponent and solutions concentrating on spheres

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Abstract

We consider the singularly perturbed coupled Schrödinger system

where \(A:=\{x\in {\mathbb {R}}^4: 0<a<|x|<b\}\) and \(\lambda _1, \lambda _2, \mu _1,\mu _2>0\). In dimension four, the Sobolev critical exponent is 4. In the cooperative case \(\beta >0\), by using a reduction approach, we prove that there exists some \(\beta _0\in (0,\sqrt{\mu _1\mu _2})\) such that system (\(P_\varepsilon \)) admits a vector solution \((u_\varepsilon ,v_\varepsilon )\) concentrating on the sphere \(|x|=a_\varepsilon \) with \(a_\varepsilon \searrow a\) as \(\varepsilon \rightarrow 0\) if \(0<\beta <\beta _0\).

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Acknowledgements

The authors would like to express their sincere gratitude to the anonymous referee for his/her valuable suggestions and comments.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, People’s Republic of China

    Jianjun Zhang

  2. Department of Mathematics, Federal University of Paraíba, João Pessoa, PB, 58051-900, Brazil

    João Marcos do Ó

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  1. Jianjun Zhang
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Correspondence to João Marcos do Ó.

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Communicated by A. Malchiodi.

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Research partially supported by INCTmat/MCT/Brazil. J. M. do Ó was supported by CNPq, CAPES/Brazil. J. J. Zhang was supported by NSFC (No. 11871123).

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Zhang, J., do Ó, J.M. Spiked vector solutions of coupled Schrödinger systems with critical exponent and solutions concentrating on spheres. Calc. Var. 58, 98 (2019). https://doi.org/10.1007/s00526-019-1540-1

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