A Brezis-Nirenberg result for non-local critical equations  in low dimension

Raffaella Servadei; Enrico Valdinoci

doi:10.3934/cpaa.2013.12.2445

Article Contents

2013, Volume 12, Issue 6: 2445-2464. Doi: 10.3934/cpaa.2013.12.2445

This issue Previous Article Oblique derivative problems for elliptic and parabolic equations Next Article Systems of singular integral equations and applications to existence of reversed flow solutions of Falkner-Skan equations

A Brezis-Nirenberg result for non-local critical equations in low dimension

Raffaella Servadei^1, and
Enrico Valdinoci^2,

1.
Dipartimento di Matematica, Università degli Studi della Calabria, Ponte Pietro Bucci 31B, I–87036 Arcavacata di Rende
2.
Università di Roma Tor Vergata, Dipartimento di Matematica, Via della Ricerca Scientifica, I-00133 Rome

Received: April 2012

Revised: January 2013

Published: November 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 49J35, 35A15, 35S15; Secondary: 47G20, 45G05.

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