A Brezis-Nirenberg result for non-local critical equations in low dimension. (English) Zbl 1302.35413

Servadei, Raffaella; Valdinoci, Enrico

Commun. Pure Appl. Anal. 12, No. 6, 2445-2464 (2013).

Summary: The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities \[ \begin{cases} (-\Delta)^s u - \lambda u = |u|^{2^*-2}u, & \text{in } \Omega \\ u = 0, & \text{in } \mathbb R^n\setminus \Omega, \end{cases} \] 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 \(\mathbb R^n\), \(n > 2s\), with Lipschitz boundary. In the recent preprints, we investigated the existence of non-trivial solutions for this problem when \(\Omega\) is an open bounded subset of \(\mathbb R^n\) with \(n \geqslant 4s\) and, in this framework, we prove some existence results.
Aim of this paper is to complete the investigation 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 H. Brézis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437–477 (1983; Zbl 0541.35029)]. 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.

MSC:

35R11	Fractional partial differential equations
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35A15	Variational methods applied to PDEs
35S15	Boundary value problems for PDEs with pseudodifferential operators
47G20	Integro-differential operators
45G05	Singular nonlinear integral equations

Keywords:

critical nonlinearities; best critical Sobolev constant; variational techniques; Mountain Pass Theorem; Linking Theorem; integrodifferential operators; fractional Laplacian