Normalized Solutions for a System of Fractional Schrödinger Equations with Linear Coupling
Keywords:
Fractional Laplacian, Schrödinger system, normalized solutions.Abstract
We study the normalized solutions of the following fractional Schrödinger system: { (−∆)su = λ1u+µ1|u|p−2u+βv
(−∆)sv = λ2v +µ2|v|q−2v +βu in RN, in RN, with prescribed mass ∫ RN u2 = a and ∫ RN v2 = b, where s ∈ (0,1), 2 < p,q ≤ 2∗ s, β ∈ R and
µ1,µ2,a,b are all positive constants. Under different assumptions on p,q and β ∈ R, we succeed to prove several existence and nonexistence results about the normalized solutions. Specifically, in the case of mass-subcritical nonlinear terms, we overcome the lack of compactness by establishing the least energy inequality and obtain the existence of the normalized solutions for any given a,b > 0
and β ∈R. While for the mass-supercritical case, we use the generalized Pohozaev equality to get the boundedness of the Palais-Smale sequence and obtain the positive normalized solution for any β >0. Finally, in the fractional Sobolev critical case i.e., p = q = 2∗ s, we give a result about the nonexistence of the positive solution.
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