Uniqueness at the critical boundary value in dimension six

Determine whether the one-dimensional profile $w_0(x_N)$ is the unique bounded positive solution of the half-space problem $-Delta v+v=v^p$ in $mathbb{R}^6_+$, with $v=c_p$ on $partial m ext{mathbb{R}^6_+}$ and $v(x',x_6)to0$ uniformly as $x_6toinfty$, for every $p>1$.

Background

The paper studies the threshold boundary value c=cpc=c_p, where cp=w0(0)c_p=w_0(0) and w0w_0 is the positive homoclinic solution of the one-dimensional equation −w′′+w=wp-w''+w=w^p. The authors prove uniqueness of the bounded positive solution for dimensions 2≤N≤52\le N\le5 and construct non-one-dimensional solutions for N≥8N\ge8.

Dimension N=6N=6 is the intermediate case: the projected amplitude satisfies a quadratic Lane–Emden-type relation in boundary dimension five. Positive supersolutions of the corresponding inequality exist, while the pure quadratic equation has no positive entire solution, and cubic corrections can alter the behavior. Consequently, the scalar projection argument does not determine whether non-one-dimensional half-space solutions exist in this dimension.

References

For $N=6$ and $p>1$, is $w_0(x_N)$ the unique bounded positive solution of eq_Pc with $c=c_p$? We leave Problem \ref{prob_N6} open.

eq_Pc:

{−Δv+v=∣v∣p−1vin R+N,v=con ∂R+N,lim⁡xN→∞v(x′,xN)=0uniformly in x′∈RN−1,\begin{cases} -\Delta v+v=|v|^{p-1}v &\text{in }\mathbb{R}^N_+,\\ v=c &\text{on }\partial\mathbb{R}^N_+,\\ \displaystyle\lim_{x_N\to\infty}v(x',x_N)=0 &\text{uniformly in }x'\in\mathbb{R}^{N-1}, \end{cases}

— The critical boundary value for a nonlinear Schrödinger equation in a half-space: rigidity and dimensional transition  (2610.02790 - Le, 2 Oct 2026) in Problem 1 (Section 1, following equation (1.13))