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

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 $λ,\:μ>0$ 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 $ϕu$. With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for $λ$ large and $μ$ small in the case $2<p<4$. Then we obtain the nonexistence of nontrivial solutions for $λ$ large and $μ$ 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 $λ\rightarrow \infty$ and $μ\rightarrow 0$.

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