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The Brezis-Nirenberg result for the fractional Laplacian

Authors: Raffaella Servadei and Enrico Valdinoci
Journal: Trans. Amer. Math. Soc. 367 (2015), 67-102
MSC (2010): Primary 49J35, 35A15, 35S15; Secondary 47G20, 45G05
Published electronically: September 22, 2014
MathSciNet review: 3271254
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Abstract: The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation

$\displaystyle \left \{ \begin {array}{ll} (-\Delta )^s u-\lambda u=\vert u\vert... ...,\\ u=0 & {\mbox { in }} \mathbb{R}^n\setminus \Omega \,, \end{array} \right . $

where $ (-\Delta )^s$ is the fractional Laplace operator, $ s\in (0,1)$, $ \Omega $ is an open bounded set of $ \mathbb{R}^n$, $ n>2s$, with Lipschitz boundary, $ \lambda >0$ is a real parameter and $ 2^*=2n/(n-2s)$ is a fractional critical Sobolev exponent.

In this paper we first study the problem in a general framework; indeed we consider the equation

$\displaystyle \left \{ \begin {array}{ll} \mathcal L_K u+\lambda u+\vert u\vert... ...mega ,\\ u=0 & \mbox {in } \mathbb{R}^n\setminus \Omega \,, \end{array}\right .$

where $ \mathcal L_K$ is a general non-local integrodifferential operator of order $ s$ and $ f$ is a lower order perturbation of the critical power $ \vert u\vert^{2^*-2}u$. In this setting we prove an existence result through variational techniques.

Then, as a concrete example, we derive a Brezis-Nirenberg type result for our model equation; that is, we show that if  $ \lambda _{1,s}$ is the first eigenvalue of the non-local operator  $ (-\Delta )^s$ with homogeneous Dirichlet boundary datum, then for any  $ \lambda \in (0, \lambda _{1,s})$ there exists a non-trivial solution of the above model equation, provided $ n\geq 4s$. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators.

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Additional Information

Raffaella Servadei
Affiliation: Dipartimento di Matematica e Informatica, Università della Calabria, Ponte Pietro Bucci 31 B, 87036 Arcavacata di Rende, Cosenza, Italy

Enrico Valdinoci
Affiliation: Dipartimento di Matematica, Università di Milano, Via Cesare Saldini 50, 20133 Milano, Italy and Weierstrass Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, D-10117 Berlin, Germany and Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy

Keywords: Mountain Pass Theorem, critical non-linearities, best critical Sobolev constant, variational techniques, integrodifferential operators, fractional Laplacian
Received by editor(s): December 16, 2011
Received by editor(s) in revised form: May 29, 2012
Published electronically: September 22, 2014
Additional Notes: The first author was supported by the MIUR National Research Project Variational and Topological Methods in the Study of Nonlinear Phenomena, and the second author by the ERC grant $𝜀$ (Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities) and the FIRB project A&B (Analysis and Beyond).
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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