### Abstract

In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator L_{K} (Formula Presented), that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445–2464, 2013). Here s∈(0,1),Ω is an open bounded set of R^{n},n>2s, with continuous boundary, λ is a positive real parameter, 2^{∗}=2n/(n-2s) is a fractional critical Sobolev exponent and f is a lower order perturbation of the critical power |u|^{2∗-2}u, while L_{K} is the integrodifferential operator defined as (Formula Presented). Under suitable growth condition on f, we show that this problem admits non-trivial solutions for any positive parameter λ. This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when K(x)=|x|^{-(n+2s)} (this gives rise to the fractional Laplace operator -(-^{Δ)s}), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities.

Original language | English |
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Pages (from-to) | 655-676 |

Number of pages | 22 |

Journal | Revista Matematica Complutense |

Volume | 28 |

Issue number | 3 |

DOIs | |

Publication status | Published - 12 Sep 2015 |

Externally published | Yes |

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### Cite this

*Revista Matematica Complutense*,

*28*(3), 655-676. https://doi.org/10.1007/s13163-015-0170-1