论文标题
具有双电势的Schrödinger-Poisson系统的局部淋巴结溶液的无限序列
An infinite sequence of localized nodal solutions for Schrödinger-Poisson system with double potentials
论文作者
论文摘要
在本文中，我们研究了以下非线性schrödinger-poisson System \ begin {equation*} \ begin {cases} - \ varepsilon^2ΔU++++v（x） -\varepsilon^2 Δϕ=u^2,&\text{in}~ \mathbb{R}^3, \end{cases} \end{equation*} where $\varepsilon>0$ is small parameters, the linear potential $V$ and nonlinear potential $K$ are bounded and bounded away from zero.通过使用惩罚方法以及不变的下降流的方法，我们建立了无限的局部签名解决方案的无限序列，这些局部签名解决方案是由对称山间传播定理的最小值表征给出的较高拓扑类型的解决方案，并且我们确定了这些标记解决方案的浓度位置。对于单个潜力，即线性潜在$ v $或非线性潜在$ k $是一个正常数，我们证明这些局部签名的解决方案集中在局部最小值$ v $或本地最大值$ k $的本地最小值附近。此外，我们的方法适用于以下非线性schrödinger方程\ begin {equation*} - \ varepsilon^2Δu+v（x）u = k（x）f（x）f（u），〜\ text {in}〜\ mathbb {r}结果概括了陈和王（Calc.Var.Partial微分方程56：1-26，2017）。
In this paper, we study the existence of localized sign-changing (or nodal) solutions for the following nonlinear Schrödinger-Poisson system \begin{equation*} \begin{cases} -\varepsilon^2 Δu+V(x)u+ϕu=K(x)f(u),&\text{in}~\mathbb{R}^3,\\ -\varepsilon^2 Δϕ=u^2,&\text{in}~ \mathbb{R}^3, \end{cases} \end{equation*} where $\varepsilon>0$ is small parameters, the linear potential $V$ and nonlinear potential $K$ are bounded and bounded away from zero. By using the penalization method together with the method of invariant sets of descending flow, we establish the existence of an infinite sequence of localized sign-changing solutions which are higher topological type solutions given by the minimax characterization of the symmetric mountain pass theorem and we determine a concrete set as the concentration position of these sign-changing solutions. For single potential, that is, linear potential $V$ or nonlinear potential $K$ is a positive constant, we prove that these localized sign-changing solutions concentrated near a local minimum set of the potential $V$ or a local maximum set of the potential $K$. Moreover, our method is works for the following nonlinear Schrödinger equation \begin{equation*} -\varepsilon^2 Δu+V(x)u=K(x)f(u),~\text{in}~\mathbb{R}^N \end{equation*} where $N\geq 2$. The result generalizes the result by Chen and Wang (Calc.Var.Partial Differential Equations 56:1-26, 2017).
