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.

A Brezis-Nirenberg result for non-local critical equations in low dimension / R. Servadei, E. Valdinoci. - In: COMMUNICATIONS ON PURE AND APPLIED ANALYSIS. - ISSN 1534-0392. - 12:6(2013), pp. 2445-2464. [10.3934/cpaa.2013.12.2445]

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

E. Valdinoci
Ultimo
2013

Abstract

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.
Best critical Sobolev constant; Critical nonlinearities; Fractional Laplacian; Integrodifferential operators; Linking theorem; Mountain pass theorem; Variational techniques
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/248765
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