Iula, Stefano and Maalaoui, Ali and Martinazzi, Luca.
(2016)
* A fractional Moser-Trudinger type inequalitiy in one dimension and its critical points.*
Differential and Integral Equations, 29 (5/6).
pp. 455-492.

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## Abstract

We show a sharp fractional Moser-Trudinger type inequality in dimension 1, i.e., for any interval I⋐R and p∈(1,∞) there exists αp>0 such that

supu∈H~1p,p(I):∥(−Δ)12pu∥Lp(I)≤1∫Ieαp|u|pp−1dx=Cp|I|,

and αp is optimal in the sense that

supu∈H~1p,p(I):∥(−Δ)12pu∥Lp(I)≤1∫Ih(u)eαp|u|pp−1dx=+∞,

for any function h:[0,∞)→[0,∞) with limt→∞h(t)=∞. Here, H~1p,p(I)={u∈Lp(R):(−Δ)12pu∈Lp(R),supp(u)⊂I¯}. Restricting ourselves to the case p=2, we further consider for λ>0 the functional

J(u):=12∫R|(−Δ)14u|2dx−λ∫I(e12u2−1)dx,u∈H~12,2(I),

and prove that it satisfies the Palais-Smale condition at any level c∈(−∞,π). We use these results to show that the equation

(−Δ)12u=λue12u2in I,

has a positive solution in H~12,2(I) if and only if λ∈(0,λ1(I)), where λ1(I) is the first eigenvalue of (−Δ)12 on I. This extends to the fractional case for some previous results proven by Adimurthi for the Laplacian and the p-Laplacian operators. Finally, with a technique by Ruf, we show a fractional Moser-Trudinger inequality on R.

supu∈H~1p,p(I):∥(−Δ)12pu∥Lp(I)≤1∫Ieαp|u|pp−1dx=Cp|I|,

and αp is optimal in the sense that

supu∈H~1p,p(I):∥(−Δ)12pu∥Lp(I)≤1∫Ih(u)eαp|u|pp−1dx=+∞,

for any function h:[0,∞)→[0,∞) with limt→∞h(t)=∞. Here, H~1p,p(I)={u∈Lp(R):(−Δ)12pu∈Lp(R),supp(u)⊂I¯}. Restricting ourselves to the case p=2, we further consider for λ>0 the functional

J(u):=12∫R|(−Δ)14u|2dx−λ∫I(e12u2−1)dx,u∈H~12,2(I),

and prove that it satisfies the Palais-Smale condition at any level c∈(−∞,π). We use these results to show that the equation

(−Δ)12u=λue12u2in I,

has a positive solution in H~12,2(I) if and only if λ∈(0,λ1(I)), where λ1(I) is the first eigenvalue of (−Δ)12 on I. This extends to the fractional case for some previous results proven by Adimurthi for the Laplacian and the p-Laplacian operators. Finally, with a technique by Ruf, we show a fractional Moser-Trudinger inequality on R.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Analysis (Martinazzi) |
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UniBasel Contributors: | Martinazzi, Luca and Iula, Stefano and Maalaoui, Ali |

Item Type: | Article, refereed |

Article Subtype: | Research Article |

Publisher: | Khayyam Publishing |

ISSN: | 0893-4983 |

Note: | Publication type according to Uni Basel Research Database: Journal article |

Language: | English |

Related URLs: | |

Last Modified: | 30 Aug 2016 14:45 |

Deposited On: | 30 Aug 2016 14:40 |

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