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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, (Pɛ): ɛ2∆v(x)-V(x)v(x) + |v(x)|p-1v(x) = 0, x ∈ ℝN, and v(x) → 0 as |x| → +∞, for the infinite case as described by Byeon and Wang. Critical means that 0 ≤ V ∈ C(ℝN) verifies Ƶ = {V = 0} ≠ ∅. Infinite means that Ƶ = {x0} and that, grossly speaking, the potential V decays at an exponential rate as x → x0. For the semiclassical limit, ɛ → 0, the infinite case has a characteristic limit problem, (Pinf): ∆u(x)-P(x)u(x) + |u(x)|p-1u(x) = 0, x ∈ Ω, with u(x) = 0 as x ∈ Ω, where Ω ⊆ ℝN 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 vk,ɛ, a solution of (Pɛ), subconverges, up to a scaling, to a corresponding solution of (Pinf ), and that vk,ɛ exponentially decays out of Ω. Finally, uniform estimates on ∂Ω for scaled solutions of (Pɛ) are obtained.