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