Article Contents
Article Contents

# Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials

• *Corresponding author: Jun Wang

This work was supported by NNSF of China(Grants 11971202), Outstanding Young foundation of Jiangsu Province No. BK20200042 and the Six big talent peaks project in Jiangsu Province(XYDXX-015)

• In the present paper we study a class of Schrödinger-Poisson equations

$$$\begin{cases} -\Delta u+V(x)u+\phi u = a (x)|u|^{p-1}u,\ x\in\mathbb{R}^3\\ -\Delta \phi = u^{2},\ x\in\mathbb{R}^3, \end{cases}$$\quad\quad\quad (1)$

where $V(x)$ and $a(x)$ are of different forms on the half space, i.e. $V(x) = V_{1}(x), a(x) = a_{1}(x)$ for $x_{1}>0$ and $V(x) = V_{2}(x), a(x) = a_{2}(x)$ for $x_{1}<0$, where $V_{1},V_{2},a_{1}$ and $a_{2}$ are periodic in each coordinate direction. By using a concentration compactness discussion, we establish the existence of surface gap soliton ground state of (1) for $p\in [3,5)$. We also give a Mountain-Pass type ground state of (1) for $p\in (3,5)$.

Mathematics Subject Classification: Primary: 35J61, 35J20, 35Q55; Secondary: 49J40.

 Citation:

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