发表机构
Faculty of Economic Mathematics, University of Economics and Law; Vietnam National University（经济数学学院，经济与法律大学; 越南国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究半空间中非线性薛定谔方程的临界边界值问题，证明解的唯一性依赖于维度：低维（2≤N≤5）唯一，高维（N≥8）非唯一，临界维度N=7在特定条件下非唯一，并给出完整证明方法。
AI 中文摘要
设 $w_0$ 为 $-w''+w=w^p$ 的正同宿解，并令 $c_p=w_0(0)$。我们研究如下方程的有界正解：在 $\mathbb R^N_+$ 中 $-\Delta v+v=v^p$，在 $\partial\mathbb R^N_+$ 上 $v=c_p$，且当 $x_N\to\infty$ 时 $v(x',x_N)\to0$ 一致成立。这是 Fernández 和 Weth [Math. Ann. \textbf{383} (2022), 361--397] 留下的阈值边界值问题。我们证明了依赖于维度的刚性/非唯一性图景。对于每个 $p>1$，当 $2\le N\le5$ 时，$w_0(x_N)$ 是唯一的有界正解；而当 $N\ge8$ 时，存在一个单参数族的有界非一维正解。在临界维度 $N=7$ 中，当精确约化非线性项的立方系数 $\kappa(p)$ 为正时，非唯一性成立；特别地，对所有 $p\ge2$ 有 $\kappa(p)>0$，且连续性将该范围延拓至 $2$ 以下。相应的振幅具有 Fowler 末端，其极限能量、颈部尺寸、凸起尺寸和对数周期由首阶项确定。$N=6$ 的情形以及 $N=7$ 中的互补范围仍然开放。证明结合了边界适应的 Modica 估计、向一维线性化算子核的投影、自动尾部非集中估计以及产生精确约化非线性项的非线性胞元问题。对于 $N\ge8$，约化线性化在加权空间中可逆。在 $N=7$ 中，通过非线性求解投影径向方程并利用 Schauder--Tychonoff 不动点闭合横向方程，从而绕过临界共振。
英文摘要
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 \[ -Δv+v=v^p\quad\text{in }\mathbb R^N_+, \qquad v=c_p\quad\text{on }\partial\mathbb R^N_+, \qquad v(x',x_N)\to0 \] uniformly as $x_N\to\infty$. 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 $2\le N\le5$, whereas for $N\ge8$ 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 $p\ge2$, 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 $N\ge8$ 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.
Comments38 pages
