Bound and ground states for a class of Schrödinger–Poisson systems
Bound and ground states for a class of Schrödinger–Poisson systems
We are concerned with the following Schrödinger–Poisson system: $$ \textstyle\begin{cases} -\Delta u+u+K(x)\phi u=a(x)u^{3},& x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, & x\in \mathbb{R}^{3}. \end{cases} $$ Assuming that $K(x)$ and $a(x)$ are nonnegative functions satisfying $$ \lim_{|x|\rightarrow \infty }a(x)=a_{\infty }>0, \qquad \lim _{|x|\rightarrow \infty }K(x)=0, $$ and other suitable conditions, we show …