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

We consider a nonlinear Schrödinger equation with critical frequency, (P_𝜀) : 𝜀²∆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 → x₀. For the semiclassical limit, 𝜀 → 0, the infinite case has a characteristic limit problem, (P_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_k,𝜀, a solution of (P_𝜀), subconverges, up to a scaling, to a corresponding solution of (P_inf ), and that v_k,𝜀 exponentially decays out of Ω. Finally, uniform estimates on ∂Ω for scaled solutions of (P_𝜀) are obtained.