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• W2944382938 abstract "In this paper, we study the existence and stability/instability of standing waves with a prescribed L2-norm for the following Hartree equation(0.1)iut+Δu−a|x|2u+(|x|−λ⁎|u|2)u=0,(t,x)∈R+×RN, 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)−Δu+a|x|2u+ωu−(|x|−λ⁎|u|2)u=0,x∈RN. Two cases are considered: (i) the system is free (a=0) and (ii) a harmonic trapping potential is added in the system (a>0). In case (i), by constructing a suitable submanifold of a L2-sphere, we prove the existence of a normalized solution for problem (0.2) with least energy in the L2-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 (ii), 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 (ii), we analyze the asymptotic behavior of the two solutions to problem (0.2) obtained in case (ii) as the trapping frequency a vanishes." @default.
• W2944382938 created "2019-05-16" @default.
• W2944382938 creator A5076271187 @default.
• W2944382938 date "2019-09-01" @default.
• W2944382938 modified "2023-10-13" @default.
• W2944382938 title "Normalized standing waves for the Hartree equations" @default.
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• W2944382938 doi "https://doi.org/10.1016/j.jde.2019.05.009" @default.
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