论文标题
涉及分数p-laplacian的奇异椭圆问题的存在结果
Existence results for singular elliptic problem involving a fractional p-Laplacian
论文作者
论文摘要
在本文中，要研究的问题如下 \ leqnomode \ begin {equation*} \ label {p} \ left \ {\ begin {array} {ll}（-Δ）_p^s u \ pm \ dfrac {| U |^{p-2} u} u = 0＆\ Quad \ mbox {on} \ \ \ \ m} \ end {array} \正确的。 \ end {equation*} \ reqnomode 其中$ω$是$ \ mathbb {r}^n（n \ geq 2）$包含原点的有限常规域 $ f：ω\ times \ mathbb {r} \ longrightArrow \ mathbb {r} $是一种满足适当生长条件的carathéodory函数，$（δ）_p^s $是分数p-laplacian，被定义为定义的p-laplacian $（δ）_ {p}^{s} u（x）= \ displayStyle 2 \ lim _ {\ varepsilon \ rightarrow 0} \ int _ {\ int _ {\ mathbb {r}^n \ setminus b _ { \ vert^{p-2}（u（x）-u（y））}} {\ vert x-y \ vert^{n+sp}}〜dy，~~~~ x \ in \ in \ mathbb {r}^n，$$ 其中$ b _ {\ varepsilon}（x）$是中心$ x $ and radius $ \ varepsilon $的开放$ \ varepsilon $。使用将临界点理论结合到分数强不平等的临界点理论，我们表明问题$（p _+）$至少承认了两种不同的非平凡弱解决方案。对于问题$（p _-），$我们使用浓度 - 紧凑性原理作为分数SOBOLEV空间给出弱的较低的半持续性结果，并证明该问题$（p _-）$至少可以接受一个非平凡的弱解决方案。
In this article, the problems to be studied are the following \leqnomode \begin{equation*} \label{p} \left\{\begin{array}{ll} (-Δ)_p^s u \pm \dfrac{|u|^{p-2}u}{|x|^{sp}} = λf(x,u) & \quad \mbox{in }\ Ω\\[0.3cm] u= 0 & \quad \mbox{on }\ \mathbb{R}^N \setminus Ω,\tag{P$_{\pm}$} \end{array} \right. \end{equation*} \reqnomode where $Ω$ is a bounded regular domain in $\mathbb{R}^N(N\geq 2)$ containing the origin, $p>1$, $s\in(0,1)$, $(N>ps)$, $λ>0$, $f : Ω\times \mathbb{R} \longrightarrow \mathbb{R}$ is a Carathéodory function satisfying a suitable growth condition and $(-Δ)_p^s$ is the fractional p-Laplacian defined as $$(-Δ)_{p}^{s} u(x) = \displaystyle 2 \lim_{\varepsilon \rightarrow 0} \int_{\mathbb{R}^N \setminus B_{\varepsilon}(x)} \dfrac{\vert u(x)-u(y) \vert^{p-2}(u(x)-u(y))}{\vert x-y \vert^{N+sp}} ~dy, ~~~~ x \in \mathbb{R}^N,$$ where $B_{\varepsilon}(x)$ is the open $\varepsilon$-ball of centre $x$ and radius $\varepsilon$. Using the critical point theory combining to the fractional Hardy inequality, we show that the problem $(P_+)$ admits at least two distinct nontrivial weak solutions. For the problem $(P_-),$ we use the concentration-compactness principle for fractional Sobolev spaces to give a weak lower semicontinuity result and prove that problem $(P_-)$ admits at least one non-trivial weak solution.
