Normalized solutions to the fractional Schrodinger equations with combined nonlinearities | |
Luo, Haijun1; Zhang, Zhitao2,3 | |
2020-08-13 | |
Source Publication | CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS |
ISSN | 0944-2669 |
Volume | 59Issue:4Pages:35 |
Abstract | We 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. |
DOI | 10.1007/s00526-020-01814-5 |
Indexed By | SCI |
Language | 英语 |
Funding Project | Fundamental Research Funds for the Central Universities[531118010205] ; National Natural Science Foundation of China[11771428] ; National Natural Science Foundation of China[11926335] |
WOS Research Area | Mathematics |
WOS Subject | Mathematics, Applied ; Mathematics |
WOS ID | WOS:000562758200002 |
Publisher | SPRINGER HEIDELBERG |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/52022 |
Collection | 数学所 |
Corresponding Author | Zhang, Zhitao |
Affiliation | 1.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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