The critical boundary value for a nonlinear Schrödinger equation in a half-space: rigidity and dimensional transition
Abstract
Let w0 be the positive homoclinic solution of −w''+w=wp and set cp=w0(0). We study bounded positive solutions of −Δv+v=vp in ℝN+, v=cp on ∂ℝN+, v(x',xN)→0 uniformly as xN→∞. This is the threshold boundary value left open by Fernández and Weth [Math.\ Ann.\ \textbf{383} (2022), 361--397]. We prove a dimension-dependent rigidity/nonuniqueness picture. For every p>1, the profile w0(xN) is the unique bounded positive solution when 2≤ N≤5, whereas for N≥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 κ(p) of the exact reduced nonlinearity is positive; in particular κ(p)>0 for all p≥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≥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.
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BibTeX
@misc{le2026critical,
title = {{The critical boundary value for a nonlinear Schrödinger equation in a half-space: rigidity and dimensional transition}},
author = {Phuong Le},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02790},
eprint = {2610.02790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02790},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The critical boundary value for a nonlinear Schrödinger equation in a half-space: rigidity and dimensional transition
AU - Phuong Le
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02790
UR - https://arxiv.org/abs/2610.02790
AB - Let w_0 be the positive homoclinic solution of −w''+w=w^p and set c_p=w_0(0). We study bounded positive solutions of −Δv+v=v^p in ℝ^N_+, v=c_p on ∂ℝ^N_+, v(x',x_N)→0 uniformly as x_N→∞. This is the threshold boundary value left open by Fernández and Weth [Math.\ Ann.\ \textbf{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≤ N≤5, whereas for N≥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 κ(p) of the exact reduced nonlinearity is positive; in particular κ(p)>0 for all p≥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≥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.
ER -