论文标题
半线性椭圆方程涉及功率非线性和具有边界奇异性的耐力电势
Semilinear elliptic equations involving power nonlinearities and hardy potentials with boundary singularities
论文作者
论文摘要
令$ c^2 $有界域和$ c^2 $有界域和$ c^\ subset \partialΩ$为$ c^2 $ compact submanifold无边界，dimension $ k $，$ k $，$ 0 \ leq k \ leq n-1 $。我们假设$σ= \ {0 \} $如果$ k = 0 $和$σ= \partialΩ$如果$ k = n-1 $。表示$d_σ（x）= \ mathrm {dist}（x，σ）$，然后put $l_μ=δ+μd_σ^{ - 2} $，其中$μ$是一个参数。在本文中，我们研究方程式的边界价值问题$-L_μU\ pm | u |^{p-1} u = 0 $ in $ω$，带有规定条件的条件$ u =ν$ on $ \ partialω$，其中$ p> 1 $和$ν$是$ \ partialω$的给定度量。非线性$ | u |^{p-1} u $被称为\ textit {subsoragent}或\ textIt {source}，具体取决于是否出现加号或减号。问题的独特特征的特征在于$σ$的浓度，非线性类型，指数$ p $和参数$μ$之间的相互作用。吸收案例和源案例在几个方面截然不同，因此需要完全不同的方法。在每种情况下，我们都以适当的能力来建立各种必要和充分的条件。与文献中的相关作品相比，通过进行了精细的分析，我们能够处理指数$ p $的超临界范围，而参数$μ$的关键案例证明了我们论文的新颖性。
Let $Ω\subset\mathbb{R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $Σ\subset \partialΩ$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $Σ= \{0\}$ if $k = 0$ and $Σ=\partialΩ$ if $k=N-1$. Denote $d_Σ(x)=\mathrm{dist}(x,Σ)$ and put $L_μ=Δ+ μd_Σ^{-2}$ where $μ$ is a parameter. In this paper, we study boundary value problems for equations $-L_μu \pm |u|^{p-1}u = 0$ in $Ω$ with prescribed condition $u=ν$ on $\partial Ω$, where $p>1$ and $ν$ is a given measure on $\partial Ω$. The nonlinearity $|u|^{p-1}u$ is referred to as \textit{absorption} or \textit{source} depending whether the plus sign or minus sign appears. The distinctive feature of the problems is characterized by the interplay between the concentration of $Σ$, the type of nonlinearity, the exponent $p$ and the parameter $μ$. The absorption case and the source case are sharply different in several aspects and hence require completely different approaches. In each case, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities. In comparison with related works in the literature, by employing a fine analysis, we are able to treat the supercritical ranges for the exponent $p$, and the critical case for the parameter $μ$, which justifies the novelty of our paper.
