The yamabe equation in a non-local setting. (English) Zbl 1273.49011

Summary: Aim of this paper is to study the following elliptic equation driven by a general non-local integro-differential operator \(\mathcal L_K\) such that \(\mathcal L_K u + \lambda u + |u|^{2^\ast - 2}u = 0\) in \(\Omega, u = 0\) in \(\mathbb R^n \setminus \Omega\), where \(s \in (0, 1), \Omega\) is an open bounded subset of \(\mathbb R^{n}, n > 2s\), with Lipschitz boundary, \(\lambda\) is a positive real parameter, \(2^\ast = 2n/(n - 2s)\) is a fractional critical Sobolev exponent, while \(\mathcal L_K\) is the non-local integrodifferential operator \[ \mathcal L_K u(x) = \int_{\mathbb R^n} \left( u(x + y) + u(x - y) - 2u(x)\right) K(y)dy, \quad x \in \mathbb R^n. \] As a concrete example, we consider the case when \(K(x) = |x|^{-(n + 2s)}\), which gives rise to the fractional Laplace operator \(-(-\Delta)^s\). We show that our problem admits a nontrivial solution for any \(\lambda > 0\), provided \(n\geq 4s\) and \(\lambda\) is different from the eigenvalues of \((-\Delta)^s\). 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.


49J35 Existence of solutions for minimax problems
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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