Normalized standing waves for the Hartree equations
By Xiaoyutao Luo. Journal of Differential Equations, 2019
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Abstract In this paper, we study the existence and stability/instability of standing waves with a prescribed L 2 -norm for the following Hartree equation (0.1) i u t + Δ u − a | x | 2 u + ( | x | − λ ⁎ | u | 2 ) u = 0 , ( t , x ) ∈ R + × R N , where t is time, N ≥ 3 , λ ∈ ( 0 , N ) and a ≥ 0 is a constant. To get such solutions, we look for normalized solutions of the associated stationary equation (0.2) − Δ u + a | x | 2 u + ω u − ( | x | − λ ⁎ | u | 2 ) u = 0 , x ∈ R N . Two cases are considered: (i) the system is free ( a = 0 ) and ( i i ) a harmonic trapping potential is added in the system ( a > 0 ) . In case (i), by constructing a suitable submanifold of a L 2 -sphere, we prove the existence of a normalized solution for problem (0.2) with least energy in the L 2 -sphere, which corresponds to a normalized ground state standing wave of problem (0.1) . Then, we show that each normalized ground state of problem (0.2) coincides a ground state of problem (0.2) in the usual sense. Furthermore, we give a global existence result for problem (0.1) and also show that any normalized ground state standing wave is strongly unstable. In case ( i i ) , for small a > 0 , we prove the existence of two different standing waves, one corresponds to a topological local minimizer of the constrained energy functional, the other corresponds to a mountain pass type solution of problem (0.2) with its energy strictly larger than that of the normalized ground state. Moreover, we show that the set of normalized ground state solutions for problem (0.2) is stable under the associated flow. Finally, to connect case (i) and case ( i i ) , we analyze the asymptotic behavior of the two solutions to problem (0.2) obtained in case ( i i ) as the trapping frequency a vanishes.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/49766903c8bf3d7dd4b369f1dc999f0691795f4f
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Citations in the corpus
- Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability (2021, background): “[15] H. Luo, Nontrivial solutions for nonlinear Schrödinger Choquard equations with critical exponents, Appl.”