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Closed-form formula for q_n appearing in far-field amplitudes

Determine a closed-form expression for the constant q_n, defined by q_n = lim_{s→0} Q(s) where Q(|x|) is the positive radial ground state solution of Δu = u − |x|^{2−n} u^3 on R^3, which enters the leading-order amplitudes of localized ring and spot B solutions.

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Background

In Theorem 2.7, the leading-order amplitudes of localized ring and spot B solutions depend on q_n, defined via the small-radius limit of the radial ground state Q solving Δu = u − |x|{2−n} u3. The authors compute an explicit closed form for ν_n but not for q_n.

Although q_n can be approximated numerically (and is plotted in Figure 1), an explicit analytic formula would sharpen the asymptotics and constants in the main existence results.

References

While we are unable to derive a closed form expression for $q_n$ in the same way as for $\nu_n$, we can numerically compute its value for different choices of $n$ by solving GL:real;normal via finite difference techniques.

GL:real;normal:

(dds+n2s)2A=A−A3,A∈R,s∈[0,∞)\left(\frac{\mathrm{d}}{\mathrm{d}s} + \frac{n}{2s}\right)^{2} A = A - A^3, \qquad A\in\mathbb{R},\qquad s\in[0,\infty)

— The role of spatial dimension in the emergence of localised radial patterns from a Turing instability (2405.16927 - Hill, 27 May 2024) in After Theorem 2.7, Section 2 (Main Results)