The critical boundary value for a nonlinear Schrödinger equation in a half-space: rigidity and dimensional transition

Let $w_0$ be the positive homoclinic solution of $-w^{\mathrm{\prime }\mathrm{\prime }}+w=w^p$ and set $c_p=w_0(0)$. We study bounded positive solutions of $$-\mathrm{\Delta }v+v=v^p\text{in }{\mathbb{R}}_{+}^N,v=c_p\text{on }\mathrm{\partial }{\mathbb{R}}_{+}^N,v(x^{\mathrm{\prime }},x_N)\to 0$$ uniformly as $x_N\to \mathrm{\infty }$. This is the threshold boundary value left open by Fernández and Weth [Math.\ Ann.\ $\mathbf{\text{383}}$ (2022), 361–397]. We prove a dimension-dependent rigidity/nonuniqueness picture. For every $p>1$, the profile $w_0(x_N)$ is the unique bounded positive solution when $2\le N\le 5$, whereas for $N\ge 8$ there is a one-parameter family of bounded positive non-one-dimensional solutions. In the critical dimension $N=7$, nonuniqueness holds whenever a cubic coefficient $\kappa (p)$ of the exact reduced nonlinearity is positive; in particular $\kappa (p)>0$ for all $p\ge 2$, and continuity extends this range below $2$. The corresponding amplitudes have Fowler ends, with their limiting energy, neck size, bulge size, and logarithmic period determined to leading order. The case $N=6$, as well as the complementary range in $N=7$, remain open. The proof combines a boundary-adapted Modica estimate, projection onto the kernel of the one-dimensional linearized operator, an automatic tail non-concentration estimate, and a nonlinear cell problem yielding the exact reduced nonlinearity. For $N\ge 8$ the reduced linearization is invertible in weighted spaces. In $N=7$ the critical resonance is bypassed by solving the projected radial equation nonlinearly and closing the transverse equation by a Schauder–Tychonoff fixed point.