We consider the equation

$$\begin{aligned} \Delta _x u+u_{yy}+f(u)=0,\quad x=(x_1,\dots ,x_N)\in {{\mathbb {R}}}^N,\ y\in {{\mathbb {R}}}, \end{aligned}$$(1)

where \(N\ge 2\) and f is a sufficiently smooth function satisfying \(f(0)=0\), \(f'(0)<0\), and some natural additional conditions. We prove that equation (1) possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in \(x'=(x_1,\ldots ,x_{N-1})\) and decaying as \(|x'|\rightarrow \infty \), 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.

中文翻译：

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整个空间上椭圆方程的局部局部周期-拟周期解和相关的KAM型结果的存在

我们考虑方程式

$$ \ begin {aligned} \ Delta _x u + u_ {yy} + f（u）= 0，\ quad x =（x_1，\ dots，x_N）\ in {{\ mathbb {R}}} ^ N， \ y \ in {{\ mathbb {R}}}中，\ end {aligned} $$（1）

其中\（N \ ge 2 \）和f是满足\（f（0）= 0 \），\（f'（0）<0 \）和某些自然附加条件的足够光滑的函数。我们证明等式（1）拥有无数个正解（不考虑平移），它们在\（x'=（x_1，\ ldots，x_ {N-1}）\）中呈径向对称，并衰减为\（| x'| \ rightarrow \ infty \），在\（x_N \）中是周期性的，在y中是拟周期的。我们的分析中也包含了有关更一般方程的相关定理。我们的方法基于中心流形和KAM型结果。