Nehari Type Ground State Solutions for Asymptotically Periodic Schrödinger-Poisson Systems

Abstract

This paper is dedicated to studying the following Schrödinger-Poisson system\[ \begin{cases} -\Delta u + V(x)u + K(x) \phi(x)u = f(x,u), &x \in \mathbb{R}^{3}, \\ -\Delta \phi = K(x) u^2, &x \in \mathbb{R}^{3}, \end{cases}\]where $V(x)$, $K(x)$ and $f(x,u)$ are periodic or asymptotically periodic in $x$. We use the non-Nehari manifold approach to establish the existence of the Nehari type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions $\lim_{|t| \to \infty} \left( \int_0^t f(x,s) \, \mathrm{d}s \right)/|t|^3 = \infty$ uniformly in $x \in \mathbb{R}^3$ and\[ \left[ \frac{f(x,\tau)}{\tau^3} - \frac{f(x,t\tau)}{(t\tau)^3} \right] \operatorname{sign}(1-t) + \theta_0 V(x) \frac{|1-t^2|}{(t\tau)^2} \geq 0, \quad \forall\, x \in \mathbb{R}^3, \; t \gt 0, \; \tau \neq 0\]with constant $\theta_0 \in (0,1)$, instead of $\lim_{|t| \to \infty} \left( \int_0^t f(x,s) \, \mathrm{d}s \right)/|t|^4 = \infty$ uniformly in $x \in \mathbb{R}^3$ and the usual Nehari-type monotonic condition on $f(x,t)/|t|^3$.