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A Brezis-Nirenberg result for non-local critical equations in low dimension. (English) Zbl 06171054

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

$\left\{\begin{array}{cc}{\left(-{\Delta }\right)}^{s}u-\lambda u={|u|}^{{2}^{*}-2}u,\hfill & \text{in}{\Omega }\hfill \\ u=0,\hfill & \text{in}{ℝ}^{n}\setminus {\Omega },\hfill \end{array}\right\$

where $s\in \left(0,1\right)$ is fixed, ${\left(-{\Delta }\right)}^{s}$ 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 $n⩾4s$ 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. 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