论文标题
具有双重关键指数的Schrodinger-Poisson系统的最少能量签名解决方案的存在和渐近行为
Existence and asymptotic behavior of least energy sign-changing solutions for Schrodinger-Poisson systems with doubly critical exponents
论文作者
论文摘要
在本文中，我们关注以下具有关键非线性和关键非本地术语的Schrödinger-Poisson系统，这是由于Hardy-Little Wood-Sobolev不等式\ Begin {equation} \ begin {case} -ΔU+U+λϕ | u |^3u = | u |^4u+| u |^{q-2} u，\ \ \＆\ x \ in in \ Mathbb {r}^{3}，\\ [2mm] -Δϕ = | u |^5，\ \＆\ x \ in \ mathbb {r}^{3}，\ end {cases} \ end {equation}，其中$λ\ in \ mathbb {r} $是一个参数，而$ q \ in（2,6）$。如果$λ\ ge（\ frac {q+2} {8}）^2 $和$ q \ in（2,6）$，则上述系统没有非平凡的解决方案。如果$λ\ in（λ^*，0）$对于某些$λ^*<0 $，则向上述系统获得最小的能量径向签名解决方案$u_λ$。此外，我们将$λ$视为参数，并将$u_λ$的渐近行为分析为$λ\至0^ - $。
In this paper, we are concerned with the following Schrödinger-Poisson system with critical nonlinearity and critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality \begin{equation}\begin{cases} -Δu+u+λϕ|u|^3u =|u|^4u+ |u|^{q-2}u,\ \ &\ x \in \mathbb{R}^{3},\\[2mm] -Δϕ=|u|^5, \ \ &\ x \in \mathbb{R}^{3}, \end{cases} \end{equation} where $λ\in \mathbb{R}$ is a parameter and $q\in(2,6)$. If $λ\ge (\frac{q+2}{8})^2$ and $q\in(2,6)$, the above system has no nontrivial solution. If $λ\in (λ^*,0)$ for some $λ^*<0$, we obtain a least energy radial sign-changing solution $u_λ$ to the above system. Furthermore, we consider $λ$ as a parameter and analyze the asymptotic behavior of $u_λ$ as $λ\to 0^-$.
