Uniform lower bound for u_σ needed to control logarithmic terms in L^2

Establish a lower bound, uniform in the nonlinearity parameter σ as σ → 0, for the radially symmetric ground state u_σ of the scaled stationary nonlinear Schrödinger equation Δu + (1/σ)(|u|^{2σ} − 1)u = 0 on ℝ^d (equivalently, for the solution of the radial initial-value problem u''(r) + [(d−1)/r] u'(r) + (1/σ)(|u(r)|^{2σ} − 1)u(r) = 0 with u(0) = α(σ), u'(0) = 0), such that the term u_0 ln((1−θ)u_0 + θ u_σ) is controlled in L^2_r for θ ∈ [0,1]. This bound would enable a power-series expansion in σ of the potential V_σ and facilitate quantitative L^2_r error estimates (e.g., proving e_σ = o(σ)).

Background

In the analysis of the small-σ limit, the authors expand the ground state u_σ as u_σ = u_0 + μ0 + eσ, where u_0 is the Gausson and μ0 is a computed correction, and aim to control the remainder eσ.

To justify an L^2_r expansion of the potential V_σ and obtain quantitative error bounds such as e_σ = o(σ), one needs to handle logarithmic terms involving u_0 ln((1−θ)u_0 + θ u_σ). This requires a uniform-in-σ lower bound for u_σ that the authors were unable to derive.

Consequently, the authors proceed via a spectral-invertibility approach to obtain convergence results in H^s_r (0 ≤ s < 1) and C^∞_loc without establishing the desired L^2_r rate.