The Existence of Partially Localized Periodic–Quasiperiodic Solutions and Related KAM-Type Results for Elliptic Equations on the Entire Space

We consider the equation 1 $$\begin{aligned} \Delta _x u+u_{yy}+f(u)=0,\quad x=(x_1,\dots ,x_N)\in {{\mathbb {R}}}^N,\ y\in {R}}}, \end{aligned}$$ where $$N\ge 2$$ and f is a sufficiently smooth function satisfying $$f(0)=0$$ , $$f'(0)<0$$ some natural additional conditions. prove that (1) possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in $$x'=(x_1,\ldots ,x_{N-1})$$ decaying as $$|x'|\rightarrow \infty $$ periodic $$x_N$$ quasiperiodic y. Related theorems for more general equations included our analysis well. Our method based on center manifold KAM-type results.