Positive solutions for the Schr\"{o}dinger-Poisson system with steep potential well

In this paper, we consider the following Schr\"odinger-Poisson system \begin{equation*} \begin{cases} - \Delta u+\lambda V(x)u+ \mu\phi u=|u|^{p-2}u &\text{in $\mathbb{R}^3$},\cr -\Delta \phi=u^{2} &\text{in $\mathbb{R}^3$}, \end{cases} \end{equation*} where $\lambda,\:\mu>0$ are real parameters and $2<p<6$. Suppose that $V(x)$ represents a potential well with the bottom $V^{-1}(0)$, the system has been widely studied in the case $4\leq p<6$. In contrast, no existence result of solutions is available for the case $2<p<4$ due to the presence of the nonlocal term $\phi u$. With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for $\lambda$ large and $\mu$ small in the case $2<p<4$. Then we obtain the nonexistence of nontrivial solutions for $\lambda$ large and $\mu$ large in the case $2<p\leq3$. Finally, we explore the decay rate of the positive solutions as $|x| \rightarrow \infty$ as well as their asymptotic behavior as $\lambda \rightarrow \infty$ and $\mu \rightarrow 0$.