A Brezis-Nirenberg result for non-local critical equations in low dimension
A Brezis-Nirenberg result for non-local critical equations in low dimension
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DOI：
10.3934/cpaa.2013.12.2445
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发表时间：
2013-05
影响因子：
1
通讯作者：
Raffaella Servadei;E. Valdinoci
中科院分区：
文献类型：
--
作者：
Raffaella Servadei;E. Valdinoci
本文研究了一类涉及临界非线性的非局部分数次方程，其中$S(0，1)$是固定的，$(-\Delta)^S$是分数拉普拉斯算子，$\lambda$是正参数，$2^*$是分数Sobolev指数，$欧米茄$是$R^n，$n>；2s$的有界开子集，具有Lipschitz边界.在最近的文献[14，18，19]中，我们研究了当$Omega$是$R^n$的具有$n\geq 4s$的有界开子集时，该问题非平凡解的存在性，并在此框架下证明了一些存在性结果。本文的目的是通过考虑$2s<；n<；4s$的情况，来完成在[14，18，19]中进行的调查。在这一背景下，我们证明了我们的问题的一个存在定理，它可以被看作是低维的Brezis-Nirenberg型结果。特别地，当$S=1$(从而$n=3$)时，我们的结果是由Brezis和Nirenberg在著名文献[4]中得到的经典结果。从这个意义上讲，我们的工作可以看作是拉普拉斯算子的一些经典结果在非局部分数次算子情形的推广。
The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities \begin{eqnarray} (-\Delta)^s u-\lambda u=|u|^{2^*-2}u, in \Omega \\ u=0, in R^n\setminus \Omega, \end{eqnarray} where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive parameter, $2^*$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $R^n$, $n>2s$, with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when $\Omega$ is an open bounded subset of $R^n$ with $n\geq 4s$ and, in this framework, we prove some existence results. Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when $2s < n < 4s$. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension. In particular when $s=1$ (and consequently $n=3$) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4]. In this sense the present work may be considered as the extension of some classical results for the Laplacian to the case of non-local fractional operators.