论文标题
在半空间中的亚地带和立方非线性schrödinger方程的正界解决方案的一维对称性$ n = 4,5 $
One-dimensional symmetry of positive bounded solutions to the subcubic and cubic nonlinear Schrödinger equation in the half-space in dimensions $N=4,5$
论文作者
论文摘要
我们关注半空间dirichlet问题\ [\ left \ {\ begin {array} {ll} -ΔV+v = | v |^{p-1} v＆\ textrm {in} \ \ \ \ mathbb {r}^n_+， v = c \ \ \ \ textrm {on} \ \ partial \ mathbb {r}^n_+，＆\ lim_ {x_n \ to \ infty} v（x'，x'，x_n）= 0 \ \ \ \ \ \ \ \ \ textrm \ end {array} \ right。 \]其中$ \ mathbb {r}^n _+= \ {x \ in \ mathbb {r}^n \：\ x_n> 0 \} $对于某些$ n \ geq 2 $，而$ p> 1 $，$ p> 1 $，$ c> 0 $是常数。 Fernandez和Weth [Math]最近展示了它。安。 （2021）]在（1，\ sqrt {e}）$中存在一个明确的数字$ c_p \，仅取决于$ p $，因此，对于$ 0 <c <c_p $，有无限的有限积极解决方案，$ c_p $，对于$ c_p $，没有界面的正面解决方案。在$ C = C_P $的情况下，他们还提出了一个有趣的公开问题是否是独特的有限积极解决方案。如果$ n = 2，3 $，我们最近在[部分不同。等。应用。 （2021）]通过在低维度中的猜想证明中调整一些想法。在这里，我们首先关注$ 1 <p <3 $，并在尺寸中证明这种独特属性$ 2 \ leq n \ leq 5 $。然后，对于Cubic NLS，其中$ P = 3 $，我们以$ 2 \ leq n \ leq 4 $建立了此功能。我们的方法完全不同，依赖于表明该解决方案的Lyapunov-Schmidt类型分解启发的合适辅助功能是$ \ Mathbb {r}^r}^{n-1} $中的车道emen-emen-fowler方程的非负超溶液，对于最佳的Liouville类型结果。
We are concerned with the half-space Dirichlet problem \[\left\{\begin{array}{ll} -Δv+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+, v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\ \textrm{uniformly in}\ x'\in\mathbb{R}^{N-1}, \end{array}\right. \] where $\mathbb{R}^N_+=\{x\in \mathbb{R}^N \ : \ x_N>0\}$ for some $N\geq 2$, and $p>1$, $c>0$ are constants. It was shown recently by Fernandez and Weth [Math. Ann. (2021)] that there exists an explicit number $c_p\in (1,\sqrt{e})$, depending only on $p$, such that for $0<c<c_p$ there are infinitely many bounded positive solutions, whereas, for $c>c_p$ there are no bounded positive solutions. They also posed as an interesting open question whether the one-dimensional solution is the unique bounded positive solution in the case where $c = c_p$. If $N=2, 3$, we recently showed this one-dimensional symmetry property in [Partial Differ. Equ. Appl. (2021)] by adapting some ideas from the proof of De Giorgi's conjecture in low dimensions. Here, we first focus on the case $1<p<3$ and prove this uniqueness property in dimensions $2\leq N\leq 5$. Then, for the cubic NLS, where $p = 3$, we establish this for $2 \leq N \leq 4$. Our approach is completely different and relies on showing that a suitable auxiliary function, inspired by a Lyapunov-Schmidt type decomposition of the solution, is a nonnegative super-solution to a Lane-Emden-Fowler equation in $\mathbb{R}^{N-1}$, for which an optimal Liouville type result is available.
