SEMICLASSICAL ASYMPTOTICS OF INFINITELY MANY SOLUTIONS FOR THE INFINITE CASE OF A NONLINEAR SCHRÖDINGER EQUATION WITH CRITICAL FREQUENCY

Abstract:

We consider a nonlinear Schrödinger equation with critical frequency, (P<inf>ɛ</inf>): ɛ²∆v(x)-V(x)v(x) + |v(x)|^p-1v(x) = 0, x ∈ ℝ^N, and v(x) → 0 as |x| → +∞, for the infinite case as described by Byeon and Wang. Critical means that 0 ≤ V ∈ C(ℝ^N) verifies Ƶ = {V = 0} ≠ ∅. Infinite means that Ƶ = {x⁰} and that, grossly speaking, the potential V decays at an exponential rate as x → x<inf>0</inf>. For the semiclassical limit, ɛ → 0, the infinite case has a characteristic limit problem, (P<inf>inf</inf>): ∆u(x)-P(x)u(x) + |u(x)|^p-1u(x) = 0, x ∈ Ω, with u(x) = 0 as x ∈ Ω, where Ω ⊆ ℝ^N is a smooth bounded strictly star-shaped region related to the potential V. We prove the existence of an infinite number of solutions for both the original and the limit problem via a Ljusternik-Schnirelman scheme for even functionals. Fixed a topological level k we show that v<inf>k,ɛ</inf>, a solution of (P<inf>ɛ</inf>), subconverges, up to a scaling, to a corresponding solution of (P<inf>inf</inf> ), and that v<inf>k,ɛ</inf> exponentially decays out of Ω. Finally, uniform estimates on ∂Ω for scaled solutions of (P<inf>ɛ</inf>) are obtained.

Año de publicación:

2022

Keywords:

• Critical frequency
• Nonlinear Schrödinger equation
• Infinite case
• Semiclassical asymptotics

Fuente:

scopus