论文标题
分数Schrödinger-Poisson系统的峰溶液的边界浓度
Boundary concentration of peak solutions for fractional Schrödinger-Poisson system
论文作者
论文摘要
本文的目的是研究以下分数Schrödinger-Poisson系统的峰值解决方案的存在： \ begin {eqnarray*} \ left \ {\ arraycolsep = 1.5pt \ begin {array} {ll} \ varepsilon^{2S}（ - δ）^{s} u+u+u+ϕU = u^p，\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ω，\\ [2mm] （-Δ）^{s} ϕ = u^2，\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ω\ω，\\ [2mm] u=ϕ=0,\ \ \ \ &\ \mbox{in}\ \mathbb{R}^N\setminus Ω, \ end {array} \正确的。 \ end {eqnArray*}其中$ s \ in（0,1）$，$ n> 2s $，$ p \ in（1，\ frac {n+2s} {n+2s} {n-2s}）$，$ω$是$ \ mathbb {r} r} n $ in lipschitz in y lipschitz yrant $ ischitz和$}的界限， Laplacian操作员，$ \ Varepsilon $是一个小的正参数。通过使用Lyapunov-Schmidt减少方法，我们构造了一个单个峰解决方案$（u_ \ varepsilon，ϕ_ \ varepsilon）$，使得$ u_ \ u_ \ varepsilon $的峰位于域中，但在边界附近。为了表征解决方案的边界浓度，该解决方案以$ \ varepsilon^{2/3} $远离边界$ \partialΩ$的距离为$ \ varepsilon $趋向于0，使用了一些新的估计和分析技术。
The goal of this paper is to study the existence of peak solutions for the following fractional Schrödinger-Poisson system: \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} \varepsilon^{2s}(-Δ)^{s}u+u+ϕu=u^p,\ \ \ &\ \mbox{in}\ Ω,\\[2mm] (-Δ)^{s}ϕ=u^2,\ \ \ &\ \mbox{in}\ Ω,\\[2mm] u=ϕ=0,\ \ \ \ &\ \mbox{in}\ \mathbb{R}^N\setminus Ω, \end{array} \right. \end{eqnarray*} where $s\in(0,1)$, $N>2s$, $p\in (1,\frac{N+2s}{N-2s})$, $Ω$ is a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary, and $(-Δ)^{s}$ is the fractional Laplacian operator, $\varepsilon$ is a small positive parameter. By using the Lyapunov-Schmidt reduction method, we construct a single peak solution $(u_\varepsilon,ϕ_\varepsilon)$ such that the peak of $u_\varepsilon$ is in the domain but near the boundary. In order to characterize the boundary concentration of solutions, which concentrates at an approximate distance $\varepsilon^{2/3}$ away from the boundary $\partialΩ$ as $\varepsilon$ tends to 0, some new estimates and analytic technique are used.
