### Abstract

In this paper we study some nonlinear elliptic equations in R^{n} obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is (-Δ)su=εhu+q+u+pinRn,where s∈ (0 , 1) , n> 4 s, ε> 0 is a small parameter, p=n+2sn-2s, 0 < q< p and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov–Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case 0 < q< 1 is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

Original language | English |
---|---|

Pages (from-to) | 183-230 |

Number of pages | 48 |

Journal | Manuscripta Mathematica |

Volume | 153 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1 May 2017 |

### Fingerprint

### Cite this

^{n}.

*Manuscripta Mathematica*,

*153*(1-2), 183-230. https://doi.org/10.1007/s00229-016-0878-3

}

^{n}'

*Manuscripta Mathematica*, vol. 153, no. 1-2, pp. 183-230. https://doi.org/10.1007/s00229-016-0878-3

**Bifurcation results for a fractional elliptic equation with critical exponent in R ^{n}.** / Dipierro, Serena; Medina, María; Peral, Ireneo; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Bifurcation results for a fractional elliptic equation with critical exponent in Rn

AU - Dipierro, Serena

AU - Medina, María

AU - Peral, Ireneo

AU - Valdinoci, Enrico

PY - 2017/5/1

Y1 - 2017/5/1

N2 - In this paper we study some nonlinear elliptic equations in Rn obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is (-Δ)su=εhu+q+u+pinRn,where s∈ (0 , 1) , n> 4 s, ε> 0 is a small parameter, p=n+2sn-2s, 0 < q< p and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov–Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case 0 < q< 1 is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

AB - In this paper we study some nonlinear elliptic equations in Rn obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is (-Δ)su=εhu+q+u+pinRn,where s∈ (0 , 1) , n> 4 s, ε> 0 is a small parameter, p=n+2sn-2s, 0 < q< p and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov–Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case 0 < q< 1 is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

KW - 35B40

KW - 35D30

KW - 35J20

KW - 35R11

KW - 49N60

UR - http://www.scopus.com/inward/record.url?scp=84982151943&partnerID=8YFLogxK

U2 - 10.1007/s00229-016-0878-3

DO - 10.1007/s00229-016-0878-3

M3 - Article

VL - 153

SP - 183

EP - 230

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 1-2

ER -

^{n}. Manuscripta Mathematica. 2017 May 1;153(1-2):183-230. https://doi.org/10.1007/s00229-016-0878-3