*(English)*Zbl 06171054

Summary: The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities

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 ${\mathbb{R}}^{n}$, $n>2s$, 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 ${\mathbb{R}}^{n}$ with $n\u2a7e4s$ 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.

##### MSC:

49J35 | Minimax problems (existence) |

35A15 | Variational methods (PDE) |

35S15 | Boundary value problems for pseudodifferential operators |

47G20 | Integro-differential operators |

45G05 | Singular nonlinear integral equations |