Chain-structure solutions to a Schr\"odinger-Poisson system in $\mathbb{R}^3$

Abstract

We prove the existence of ground states and high-energy solutions to the following Schr\"odinger-Poisson system \begin{align*} \begin{cases} - \Delta u + a(x) u + u v = 0,\newline \Delta v = u^2, \end{cases} \quad \text{in } \mathbb{R}^3, \end{align*} where $a \in L^\infty(\mathbb{R}^3)$ is nonnegative and radially symmetric in the first two variables. Differing from the standard approach, our framework yields chain-structure solutions, i.e. solutions periodic in the third variable. A central part of this work is the construction of the Green function of a Poisson problem subject to periodic boundary conditions and we show that its asymptotic profile is tightly related to both the two and three dimensional Poisson problems in the entire space. If the potential $a$ is constant along the third variable, we apply symmetry techniques to construct solutions that have nonvanishing derivative in the third variable.