论文标题
半线性椭圆形Schrödinger方程涉及奇异电位和源术语
Semilinear elliptic Schrödinger equations involving singular potentials and source terms
论文作者
论文摘要
令$ c^2 $有界域，$ c^2 $界面和$ c^2 $ submanifold，$ c^2 $ submanifold无边界，$ k $，$ 0 \ leq k <n-2 $。 pot $l_μ=δ+μd_σ^{ - 2} $ in $ω\setMinusσ$，其中$d_σ（x）= \ mathrm {dist}（x，x，σ）$和$μ$是一个参数。 We study the boundary value problem (P) $-L_μu = g(u) + τ$ in $Ω\setminus Σ$ with condition $u=ν$ on $\partial Ω\cup Σ$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function and $τ$ and $ν$ are positive measures.反方势$d_σ^{ - 2} $之间的相互作用，源术语$ g（u）$的性质与度量数据$τ，ν$之间的相互作用在问题的研究中产生了很大的困难。我们基于对绿色内核和马丁内核以及适当能力引起的精细拓扑的精致估计进行深入分析，以在不同情况下为解决方案建立各种必要和充分的条件。
Let $Ω\subset \mathbb{R}^N$ ($N>2$) be a $C^2$ bounded domain and $Σ\subset Ω$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_μ= Δ+ μd_Σ^{-2}$ in $Ω\setminus Σ$, where $d_Σ(x) = \mathrm{dist}(x,Σ)$ and $μ$ is a parameter. We study the boundary value problem (P) $-L_μu = g(u) + τ$ in $Ω\setminus Σ$ with condition $u=ν$ on $\partial Ω\cup Σ$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function and $τ$ and $ν$ are positive measures. The interplay between the inverse-square potential $d_Σ^{-2}$, the nature of the source term $g(u)$ and the measure data $τ,ν$ yields substantial difficulties in the research of the problem. We perform a deep analysis based on delicate estimate on the Green kernel and Martin kernel and fine topologies induced by appropriate capacities to establish various necessary and sufficient conditions for the existence of a solution in different cases.
