论文标题
在整个空间上存在部分定位的周期性周期性验证溶液和相关的KAM型结果
The existence of partially localized periodic-quasiperiodic solutions and related KAM-type results for elliptic equations on the entire space
论文作者
论文摘要
我们考虑方程$Δ_xu+u_ {yy}+f（u）= 0，\ x =（x_1，\ dots，x_n）\ in \ in \ mathbb {r}^n，\ y \ in \ mathbb {r}，r} $ f'（0）<0 $，以及一些自然的额外条件。我们证明，该方程具有许多积极的解决方案（无视翻译），这些解决方案在$ x'=（x_1，\ dots，x_ {n-1}）$中是径向对称的，并且以$ | x'| \ to \ to \ infty $的衰减，定期为$ x_n $ in $ x_n $，以及$ y $ y $ y $ y $。我们的分析也包括了更多一般方程的相关定理。我们的方法基于中心歧管和KAM型结果。
We consider the equation $Δ_x u+u_{yy}+f(u)=0,\ x=(x_1,\dots,x_N)\in\mathbb{R}^N,\ y\in \mathbb{R},$ where $N\geq 2$ and $f$ is a sufficiently smooth function satisfying $f(0)=0$, $f'(0)<0$, and some natural additional conditions. We prove that the equation possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in $x'=(x_1,\dots,x_{N-1})$ and decaying as $|x'|\to\infty$, periodic in $x_N$, and quasiperiodic in $y$. Related theorems for more general equations are included in our analysis as well. Our method is based on center manifold and KAM-type results.
