The Yamabe equation in a non-local setting

By Raffaella Servadei. 2013

112 citations recorded by the source corpus.

Abstract. Aim of this paper is to study the following elliptic equation driven by a general non-local integrodifferential operator such that in , in , where , is an open bounded set of ℝn, , with Lipschitz boundary, λ is a positive real parameter, is a fractional critical Sobolev exponent, while is the non-local integrodifferential operator As a concrete example, we consider the case when , which gives rise to the fractional Laplace operator . In this framework, in the existence result proved along the paper, we show that our problem admits a non-trivial solution for any , provided and λ is different from the eigenvalues of . 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1b3ea846ec0152da42886829c0fefddb1d08a0e1

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In Applied Mathematics

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Citations in the corpus

On doubly nonlocal fractional elliptic equations (2015, methodology): “Indeed, the nonlocal analysis which we perform in this paper in order to use the Mountain Pass Theorem is quite general and it was successfully exploited for other goals in several recent contributions (see [16, 19, 20, 21, 23, 28, 30, 31, 32] and [15] for various properties on the fractional Sobole”
A critical fractional equation with concave-convex power nonlinearities (2013, methodology): “The linear case q 1, when the right hand side of the equation is equal to λu u s − 2 u , was treated in [37,38, 41–43 .”
On a fractional degenerate Kirchhoff-type problem (2017, background): “The reader interested in knowing more about fractional Sobolev spaces is invited to consult the very nice survey paper [19] and the references therein, while for more details on X and X0 we refer to the papers [38, 39, 40, 41, 42], where these functional spaces were introduced and analysed in their ”
BIFURCATION AND MULTIPLICITY RESULTS FOR CRITICAL NONLOCAL FRACTIONAL LAPLACIAN PROBLEMS (2016, result): “2), a positive answer has been given in the recent papers [24, 25, 30], also in presence of a perturbation of the critical term (for this see, for instance, [29, 31]): in all these papers the classical and well known existence results for (1.”
Bifurcation and multiplicity results for critical fractional p‐Laplacian problems (2014, unknown): “, [34, 31, 35, 32, 33, 2, 5, 37, 17]).”
The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem (2018, unknown): “, in the fractional setting s < 1 much less is known though much eﬀort have been dedicated very recently. Critical equation with the fractional Laplacian in bounded domains have been considered in [3, 11, 34, 33, 35, 37, 39] when p = 2 and in [27] for a general p. Concerning critical equations in”
Three Weak Solutions for Nonlocal Fractional Equations (2014, unknown): “For further details on the fractional Sobolev spaces we refer to [1] and to the references therein, while for other details on X and X0 we refer to [12], where these functional spaces were introduced, and also to [11, 13, 14, 15], where various properties of these spaces were proved.”
Higher nonlocal problems with bounded potential (2014, background): “For further details on the fractional Sobolev spaces we refer to [7] and to the references therein, while for other details on X and X0 we refer to [13], where these functional spaces were introduced, and also to [12, 14, 15, 16], where various properties of these spaces were proved.”
The Brezis–Nirenberg problem for the fractional p-Laplacian (2015, background): “5 [41] R. Servadei.”