TY - JOUR

T1 - A brezis-nirenberg result for non-local critical equations in low dimension

AU - Servadei, Raffaella

AU - Valdinoci, Enrico

PY - 2013/11/1

Y1 - 2013/11/1

N2 - The present paper is devoted to the study of the following nonlocal fractional equation involving critical nonlinearities { (-δ) ∈u -u = u2-2u in ω u = 0 in Rn n ω where s 2 (0; 1) is fixed, (-δ)s is the fractional Laplace operator, is a positive parameter, 2 is the fractional critical Sobolev exponent and is an open bounded subset of Rn, n > 2s , with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when is an open bounded subset of Rn 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 < 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.

AB - The present paper is devoted to the study of the following nonlocal fractional equation involving critical nonlinearities { (-δ) ∈u -u = u2-2u in ω u = 0 in Rn n ω where s 2 (0; 1) is fixed, (-δ)s is the fractional Laplace operator, is a positive parameter, 2 is the fractional critical Sobolev exponent and is an open bounded subset of Rn, n > 2s , with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when is an open bounded subset of Rn 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 < 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.

KW - Best critical Sobolev constant

KW - Critical nonlinearities

KW - Fractional Laplacian

KW - Integrodifferential operators

KW - Linking theorem

KW - Mountain pass theorem

KW - Variational techniques

UR - http://www.scopus.com/inward/record.url?scp=84878285487&partnerID=8YFLogxK

U2 - 10.3934/cpaa.2013.12.2445

DO - 10.3934/cpaa.2013.12.2445

M3 - Article

AN - SCOPUS:84878285487

VL - 12

SP - 2445

EP - 2464

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

SN - 1534-0392

IS - 6

ER -