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Normalized solutions to the fractional Schrodinger equations with combined nonlinearities
Luo, Haijun1; Zhang, Zhitao2,3
2020-08-13
Source PublicationCALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
ISSN0944-2669
Volume59Issue:4Pages:35
AbstractWe study the normalized solutions of the fractional nonlinear Schrodinger equations with combined nonlinearities (-Delta)(s)u = lambda u + mu vertical bar u vertical bar(q-2)u + vertical bar u vertical bar(p-2)u in R-N, and we look for solutions which satisfy prescribed mass integral(RN) vertical bar u vertical bar(2) = a(2), where N >= 2, s is an element of (0, 1), mu is an element of R and 2 < q < p < 2*(s) = 2N/(N - 2s). Under different assumptions on q < p, a > 0 and mu is an element of R, we prove some existence and nonexistence results about the normalized solutions. More specifically, in the purely L-2-subcritical case, we overcome the lack of compactness by virtue of the monotonicity of the least energy value and obtain the existence of ground state solution for mu > 0. While for the defocusing situation mu < 0, we prove the nonexistence result by constructing an auxiliary function. We emphasis that the nonexistence result is new even for Laplacian operator. In the purely L-2-supercritical case, we introduce a fiber energy functional to obtain the boundedness of the Palais-Smale sequence and get a mountain-pass type solution. In the combined-type cases, we construct different linking structures to obtain the saddle type solutions. Finally, we remark that we prove a uniqueness result for the homogeneous nonlinearity (mu = 0), which is based on the Morse index of ground state solutions.
DOI10.1007/s00526-020-01814-5
Indexed BySCI
Language英语
Funding ProjectFundamental Research Funds for the Central Universities[531118010205] ; National Natural Science Foundation of China[11771428] ; National Natural Science Foundation of China[11926335]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000562758200002
PublisherSPRINGER HEIDELBERG
Citation statistics
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/52022
Collection数学所
Corresponding AuthorZhang, Zhitao
Affiliation1.Hunan Univ, Sch Math, Changsha 410082, Hunan, Peoples R China
2.Chinese Acad Sci, Acad Math & Syst Sci, HLM, Beijing 100190, Peoples R China
3.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
Recommended Citation
GB/T 7714
Luo, Haijun,Zhang, Zhitao. Normalized solutions to the fractional Schrodinger equations with combined nonlinearities[J]. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS,2020,59(4):35.
APA Luo, Haijun,&Zhang, Zhitao.(2020).Normalized solutions to the fractional Schrodinger equations with combined nonlinearities.CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS,59(4),35.
MLA Luo, Haijun,et al."Normalized solutions to the fractional Schrodinger equations with combined nonlinearities".CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 59.4(2020):35.
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