The Yamabe equation in a non-local setting

Abstract

Aim of this paper is to study the following elliptic equation driven by a general non-local integrodifferential operator LK such that LKu + λu + |u|p-2u = 0 in Ω, u = 0 in ℝn\Ω, wheres ∈ (0, 1), Ω is an open bounded set of ℝn, n > 2s, with Lipschitz boundary, λ is a positive real parameter, 2∗ = 2n/(n - 2s) is a fractional critical Sobolev exponent, while LK is the non-local integrodifferential operator LKu(x) = fℝn (u(x + y) + u(x - y) - 2u(x))K(y) dy, x ∈ℝn. As a concrete example, we consider the case when K(x) = |x|-(n+2s), which gives rise to the fractional Laplace operator -(-Δ)s. In this framework, in the existence result proved along the paper, we show that our problem admits a non-trivial solution for any λ > 0, provided n ≥ 4s and λ is different from the eigenvalues of (-δ)s. This result may be read as the non-local fractional counterpart of the one obtained by Capozzi, Fortunato and Palmieri and by Gazzola and Ruf for the classical Laplace equation with critical nonlinearities. In this sense the present work may be seen as the extension of some classical results for the Laplacian to the case of non-local fractional operators.