论文标题
具有单数电势和吸收项的半线性椭圆形schrödinger方程
Semilinear elliptic Schrödinger equations with singular potentials and absorption terms
论文作者
论文摘要
令$ c^2 $有界域和$ c^2 $界面域，$ c^2 $有界域和$ c^2 $ submanifold，$ c^2 $ submanifold，没有边界，$ k $ dimension $ k $，$ 0 \ leq 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 investigate 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.反方势能$d_σ^{ - 2} $，吸收项$ g（u）$与测量数据$τ，ν$之间的竞争效应之间的复杂相互作用揭示了问题（p）可解决的情况。我们为存在解决方案的$ g $增长提供了鲜明的条件。当$ g $是一个电源函数时，即$ g（u）= | u |^{p-1} u $，$ p> 1 $，我们表明问题（p）承认了几个关键指数，因为在亚临界情况下存在单数解决方案，在亚临界情况下存在单数解决方案（即，$ p $在关键的指数中都比$ pribalitive）在$ pritagential Case（即$ p的范围内都要重复）。最后，我们建立了以适当的能力（p）的适当能力表示的各种必要和充分的条件。
Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) 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 investigate 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 complex interplay between the competing effects of the inverse-square potential $d_Σ^{-2}$, the absorption term $g(u)$ and the measure data $τ,ν$ discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of $g$ for the existence of solutions. When $g$ is a power function, namely $g(u)=|u|^{p-1}u$ with $p>1$, we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. $p$ is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. $p$ is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P).
