The Brezis-Nirenberg result for the fractional Laplacian

• Published in: Transactions of the American Mathematical Society 2015-01, Vol.367 (1), p.67-102
• Main Authors: SERVADEI, RAFFAELLA, VALDINOCI, ENRICO
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-2014-05884-4

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 $\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{\mathrm{s}}\mathrm{u}-\mathrm{\lambda }\mathrm{u}=|\mathrm{u}{|}^{2^{*}-{2}_{\mathrm{u}}}& \mathrm{i}\mathrm{n} \ \mathrm{\Omega },\\ \mathrm{u}=0& \mathrm{i}\mathrm{n} \ {\mathrm{\mathbb{R}}}^{\mathrm{n}}\backslash \mathrm{\Omega },\end{array}$ where (-Δ)s is the fractional Laplace operator, s ∈ (0, 1), Ω is an open bounded set of Rn, n > 2s, with Lipschitz boundary, λ > 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 $\{\begin{array}{ll}{\mathcal{L}}_{\mathrm{K}}\mathrm{u}+\mathrm{\lambda }\mathrm{u}+|\mathrm{u}{|}^{{2}^{*}-2}\mathrm{u}+\mathrm{f}(\mathrm{x},\mathrm{u})=0& \mathrm{i}\mathrm{n} \ \mathrm{\Omega },\\ \mathrm{u}=0& \mathrm{i}\mathrm{n} \ {\mathrm{\mathbb{R}}}^{\mathrm{n}}\backslash \mathrm{\Omega },\end{array}$ where LK is a general non-local integrodifferential operator of order s and f is a lower order perturbation of the critical power |u|2*–2u. 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 λ1,s is the first eigenvalue of the non-local operator (-Δ)s with homogeneous Dirichlet boundary datum, then for any λ ∈ (0, λ1,s) there exists a non-trivial solution of the above model equation, provided n ≥ 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.