论文标题
具有陡峭势井的Schrödinger-Poisson系统的阳性解决方案
Positive solutions for the Schrödinger-Poisson system with steep potential well
论文作者
论文摘要
在本文中，我们考虑以下schrödinger -poisson系统\ begin {equation*} \ begin {cases} - ΔU+λv（x）u+μxto= | u |^| |^{p -2} u＆\ text {in $ $ \ mathbb {r}^3 $}，\ end {cases} \ end {equation*}其中$λ，\：μ> 0 $是真实参数，$ 2 <p <6 $。假设$ v（x）$在底部$ v^{ - 1}（0）$中代表潜在的井，该系统已在$ 4 \ leq p <6 $的情况下进行了广泛研究。相比之下，由于存在非局部项$ ϕu $，因此没有解决方案的存在的结果$ 2 <p <4 $。借助截断技术和参数依赖性的紧凑型引理，我们首先证明存在$λ$大的阳性解决方案，而在$ 2 $ 2 <p <4 $的情况下为$λ$ $。然后，我们以$λ$大的$λ$和$μ$在$ 2 <p \ leq3 $中获得非平凡解决方案的不存在。最后，我们以$ | x |探索阳性解决方案的衰减率。 \ rightarrow \ infty $以及它们的渐近行为为$λ\ rightarrow \ infty $和$μ\ rightarrow 0 $。
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 $μ\rightarrow 0$.
