论文标题
具有两个参数的一类关键Schrödinger-Poisson系统的解决方案多样性
Multiplicity of solutions for a class of critical Schrödinger-Poisson system with two parameters
论文作者
论文摘要
我们研究形式的一类关键schrödinger -poisson系统{等式*} \ begin {cases}-Δu+λv（x）u+ϕu =μ| \ Mathbb {r}^3，\\\ end {cases} \ end {equation*}其中$λ，μ> 0 $是两个参数，$ p \ in（4,6）$和$ v $满足某些潜在的良好状况。通过使用各种论点，我们证明存在$λ$的$λ$和$μ> 0 $的正面状态解决方案，它们的渐近行为是$λ\ to \ infty $。此外，通过使用ljusternik-schnirelmann理论，如果$λ$很大并且$μ$很小，我们就会获得多个正溶液的存在。
We study a class of critical Schrödinger-Poisson system of the form \begin{equation*} \begin{cases} -Δu+λV(x)u+ϕu=μ|u|^{p-2}u+|u|^{4}u& \quad x\in \mathbb{R}^3,\\ -Δϕ=u^2&\quad x\in \mathbb{R}^3,\\ \end{cases} \end{equation*} where $λ, μ>0$ are two parameters, $p\in(4,6)$ and $V$ satisfies some potential well conditions. By using the variational arguments, we prove the existence of positive ground state solutions for $λ$ large enough and $μ>0$, and their asymptotical behavior as $λ\to\infty$. Moreover, by using Ljusternik-Schnirelmann theory, we obtain the existence of multiple positive solutions if $λ$ is large and $μ$ is small.
