论文标题
绿色操作员的紧凑性，并应用于半线性非本地椭圆方程
Compactness of Green operators with applications to semilinear nonlocal elliptic equations
论文作者
论文摘要
在本文中，我们考虑了一类Integro-Differential运算符$ \ Mathbb {l} $在$ C^2 $有限域$ω\ subset \ subset \ mathbb {r}^n $的情况下，并具有适当同质的dirichlet条件，其中每种情况都承认了一个常见的相反操作员，该竞争者通常是绿色操作员$ \ nmathb} $ \ c}。在$ \ mathbb {l} $及其绿色操作员的轻度条件下，我们建立了$ \ mathbb {g}^ω$涉及加权Lebesgue空间和加权度量空间的各种尖锐紧凑性。然后使用这些结果来证明半连续性椭圆方程$ \ MATHBB {l} u + g（u）=μ$ in $ω$，带边界条件$ u = 0 $ in $ \ partialω$或外部条件$ u = 0 $ u = 0和$ g：\ mathbb {r} \ to \ mathbb {r} $是一个不满意的连续函数，满足了亚临界的整体条件。当$ g（t）= | t |^{p-1} t $带有$ p> 1 $时，我们提供了一种以合适的贝塞尔能力来表达的鲜明条件，以解决解决方案。本文的贡献包括（i）开发新型的统一技术，这些技术允许处理各种类型的分数操作员以及（ii）获得尖锐的紧凑性和存在的鲜明的水平，从而导致加权空间，从而在文献中完善并扩展了几个相关的结果。
In this paper, we consider a class of integro-differential operators $\mathbb{L}$ posed on a $C^2$ bounded domain $Ω\subset \mathbb{R}^N$ with appropriate homogeneous Dirichlet conditions where each of which admits an inverse operator commonly known as the Green operator $\mathbb{G}^Ω$. Under mild conditions on $\mathbb{L}$ and its Green operator, we establish various sharp compactness of $\mathbb{G}^Ω$ involving weighted Lebesgue spaces and weighted measure spaces. These results are then employed to prove the solvability for semilinear elliptic equation $\mathbb{L} u + g(u) = μ$ in $Ω$ with boundary condition $u=0$ on $\partial Ω$ or exterior condition $u=0$ in $\mathbb{R}^N \setminus Ω$ if applicable, where $μ$ is a Radon measure on $Ω$ and $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing continuous function satisfying a subcriticality integral condition. When $g(t)=|t|^{p-1}t$ with $p>1$, we provide a sharp sufficient condition expressed in terms of suitable Bessel capacities for the existence of a solution. The contribution of the paper consists of (i) developing novel unified techniques which allow to treat various types of fractional operators and (ii) obtaining sharp compactness and existence results in weighted spaces, which refine and extend several related results in the literature.
