Extend blow-up existence from strict maxima to non-degenerate boundary mean-curvature critical points

Establish the existence, for sufficiently large λ, of boundary blow-up solutions concentrating at boundary points where the mean curvature H of ∂Ω has a non-degenerate (stable) critical point, rather than a strict local maximum, for both (i) the scalar Neumann problem −Δu + λu = u^3 in a smooth bounded domain Ω ⊂ R^4 with ∂νu = 0, and (ii) the two-component competitive Neumann system −Δu_i + λu_i = u_i^3 − β u_i u_j^2 in Ω with ∂νu_i = 0 for i = 1,2 and β > 0; derive a C^1 expansion of the reduced energy needed to carry out this extension of the Lyapunov–Schmidt scheme.

Background

The paper constructs boundary blow-up solutions for the critical 4D Neumann system in the competitive regime, with each component concentrating at distinct boundary points where the mean curvature H attains strict local maxima with positive values. The analysis relies on a refined ansatz (involving Bessel functions) and Lyapunov–Schmidt reduction tailored to dimension four.

The authors suggest that the strict local maximum assumption on H could potentially be weakened to allow general stable (e.g., non-degenerate) critical points, but this extension is technically demanding. They note that even in the scalar case −Δu + λu = u3 with Neumann boundary condition, the corresponding generalization remains unresolved. Achieving this likely requires a C1 expansion of the reduced energy, beyond what is developed here.

References

We expect Theorem \ref{xthmy} to hold for more general stable critical points of H (e.g., non-degenerate ones). However, this remains an open problem even for the scalar equation single-equ. From a technical standpoint, extending the result would require a C1-expansion of the reduced energy defined in e-f-11, which likely necessitates a further refinement of the ansatz.

single-equ:

{−Δu+λu=uq,u>0,in Ω,∂νu=0,on ∂Ω,\begin{cases} -\Delta u+\lambda u=u^q,\quad u>0, & \text{in }\Omega,\\ \partial_\nu u=0, & \text{on }\partial\Omega, \end{cases}

— Blowing-up solutions to a critical 4D Neumann system in a competitive regime  (2603.29329 - Guo et al., 31 Mar 2026) in Remark, end of Section 1 (Introduction)