Normalized solutions for a fourth-order Schrödinger equation with positive second-order dispersion coefficient

2019/08/07 by Xiao Luo, Tao Yang, Luo, Xiao +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1908.03079

openalex publication_date 2019/08/07 · openalex created_date 2019/08/13 · openalex updated_date 2026/07/28

Abstract

We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schrödinger equation Δ²u+μΔu-λu=|u|^p-2u, ~~~~x ∈ \R^N
under the normalized constraint ∫_ℝ^N u²=a², where N ≥ 2, a,μ > 0, 2+(8)/(N) 0 and γ_p = (N(p - 2))/(4p). Furthermore, we give some asymptotic properties of the normalized solutions to (\ref1) as μ→0⁺ and a→0⁺, respectively. In conclusion, we mainly extend the results in \citeDBon,dbJB, which deal with (\ref1), from μ≤0 to the case of μ>0, and also extend the results in \citeTJLu,Nbal, which deal with (\ref1), from L²-subcritical and L²-critical setting to L²-supercritical setting.

Citations

• Orbital stability of standing waves for some nonlinear Schr�dinger equations
• Nonlinear Schr�dinger equations and sharp interpolation estimates
• A relation between pointwise convergence of functions and convergence of functionals
• Strong instability of ground states to a fourth order Schr "odinger\n equation
• Normalized solutions to the mixed dispersion nonlinear Schr "odinger\n equation in the mass critical and supercritical regime
• Some remarks on a minimization problem associated to a fourth order nonlinear Schrödinger equation
• Orbital Stability of Standing Waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions

Cited by

• Standing waves with prescribed mass for biharmonic NLS with positive dispersion and Sobolev critical exponent

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