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Normalized solutions for scalar field equation involving multiple critical nonlinearities

  • Xiaojing Feng and Haidong Liu ORCID logo EMAIL logo
Published/Copyright: October 4, 2023

Abstract

This paper concerns the scalar field equation

- Δ u = λ u + | u | q - 2 u + a | u | 4 u + b ( I 2 | u | 5 ) | u | 3 u in  3

under the normalized constraint 3 u 2 𝑑 x = c 2 , where a , b , c > 0 , 2 < q < 10 3 and I 2 is the Riesz potential. We prove that for small prescribed mass c the above equation has a positive ground state solution and an infinite sequence of normalized solutions with negative energies tending to zero. Asymptotic properties of ground state solutions as a 0 + and as b 0 + are also studied.

MSC 2020: 35J20; 35J60

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 12271313

Award Identifier / Grant number: 12071266

Award Identifier / Grant number: 12101376

Award Identifier / Grant number: 12171204

Funding statement: Xiaojing Feng is supported by NSFC (12271313, 12071266, 12101376), Fundamental Research Program of Shanxi Province (202203021211300, 202203021211309, 20210302124528) and Shanxi Scholarship Council of China (2020–005). Haidong Liu is supported by NSFC (12171204).

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Received: 2023-07-24
Published Online: 2023-10-04
Published in Print: 2024-05-01

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