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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 ${ℒ}_{K}$ such that ${ℒ}_{K}u+\lambda u+{|u|}^{{2}^{*}-2}u=0$ in ${\Omega },u=0$ in ${ℝ}^{n}\setminus {\Omega }$, where $s\in \left(0,1\right),{\Omega }$ is an open bounded subset of ${ℝ}^{n},n>2s$, with Lipschitz boundary, $\lambda$ is a positive real parameter, ${2}^{*}=2n/\left(n-2s\right)$ is a fractional critical Sobolev exponent, while ${ℒ}_{K}$ is the non-local integrodifferential operator

${ℒ}_{K}u\left(x\right)={\int }_{{ℝ}^{n}}\left(u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)\right)K\left(y\right)dy,\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{n}·$

As a concrete example, we consider the case when $K\left(x\right)={|x|}^{-\left(n+2s\right)}$, which gives rise to the fractional Laplace operator $-{\left(-{\Delta }\right)}^{s}$. We show that our problem admits a nontrivial solution for any $\lambda >0$, provided $n\ge 4s$ and $\lambda$ is different from the eigenvalues of ${\left(-{\Delta }\right)}^{s}$. 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.

##### MSC:
 49J35 Minimax problems (existence) 35A15 Variational methods (PDE) 35S15 Boundary value problems for pseudodifferential operators 47G20 Integro-differential operators 45G05 Singular nonlinear integral equations