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- Title
Normalized solutions to the fractional Schrödinger equations with combined nonlinearities.
- Authors
Luo, Haijun; Zhang, Zhitao
- Abstract
We study the normalized solutions of the fractional nonlinear Schrödinger equations with combined nonlinearities (- Δ) s u = λ u + μ | u | q - 2 u + | u | p - 2 u in R N , <graphic href="526_2020_1814_Article_Equ35.gif"></graphic> and we look for solutions which satisfy prescribed mass ∫ R N | u | 2 = a 2 , <graphic href="526_2020_1814_Article_Equ36.gif"></graphic> where N ≥ 2 , s ∈ (0 , 1) , μ ∈ R and 2 < q < p < 2 s ∗ = 2 N / (N - 2 s) . Under different assumptions on q < p , a > 0 and μ ∈ 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 μ > 0 . While for the defocusing situation μ < 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 ( μ = 0 ), which is based on the Morse index of ground state solutions.
- Publication
Calculus of Variations & Partial Differential Equations, 2020, Vol 59, Issue 4, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-020-01814-5