Positive solutions for the Schrödinger-Poisson system with steep potential well

Abstract

In this paper, we consider the following Schrödinger-Poisson system \begin{equation*} \begin{cases} - Δu+λV(x)u+ μϕu=|u|^{p-2}u &\text{in ${\mathbb{R}}^3$},\cr -Δϕ=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 $4\le p<6$. In contrast, no existence result of solutions is available for the case $2<p<4$ due to the presence of the nonlocal term $\varphi u$. 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 $2<p\le 3$. Finally, we explore the decay rate of the positive solutions as $|x|\to \infty$ as well as their asymptotic behavior as $\lambda \to \infty$ and $\mu \to 0$.