### Abstract

The aim of this paper is to study the following problem (P) ≡{ (-Δ)^{s}u = u^{q} + u^{p} in ; u > 0 inΩ; Bsu = 0 in R^{N}/Ω with 0 < q < 1 < p, N > 2s, > 0,Ω⊂ R^{N} is a smooth bounded domain, (-Δ)^{s}u(x) = aN;s P:V: Z R^{N} u(x) -u(y)/x-y/^{N+2s} dy; aN;s is a normalizing constant, and B_{s}u = uX∑_{1} + NsuX∑_{2} : Here, ∑_{1} and ∑_{2} are open sets in R^{N}/Ω such that∑_{2} ∩ ∑_{2} = ; and ∑_{1} ∩ ∑_{2} = R^{N}/Ω: In this setting, Nsu can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [20], and Bsu is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem (P) for suitable ranges of and p and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

Original language | English |
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Pages (from-to) | 1103-1120 |

Number of pages | 18 |

Journal | Communications on Pure and Applied Analysis |

Volume | 17 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 May 2018 |

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*Communications on Pure and Applied Analysis*,

*17*(3), 1103-1120. https://doi.org/10.3934/cpaa.2018053

}

*Communications on Pure and Applied Analysis*, vol. 17, no. 3, pp. 1103-1120. https://doi.org/10.3934/cpaa.2018053

**A nonlocal concave-convex problem with nonlocal mixed boundary data.** / Abdellaoui, Boumediene; Dieb, Abdelrazek; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A nonlocal concave-convex problem with nonlocal mixed boundary data

AU - Abdellaoui, Boumediene

AU - Dieb, Abdelrazek

AU - Valdinoci, Enrico

PY - 2018/5/1

Y1 - 2018/5/1

N2 - The aim of this paper is to study the following problem (P) ≡{ (-Δ)su = uq + up in ; u > 0 inΩ; Bsu = 0 in RN/Ω with 0 < q < 1 < p, N > 2s, > 0,Ω⊂ RN is a smooth bounded domain, (-Δ)su(x) = aN;s P:V: Z RN u(x) -u(y)/x-y/N+2s dy; aN;s is a normalizing constant, and Bsu = uX∑1 + NsuX∑2 : Here, ∑1 and ∑2 are open sets in RN/Ω such that∑2 ∩ ∑2 = ; and ∑1 ∩ ∑2 = RN/Ω: In this setting, Nsu can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [20], and Bsu is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem (P) for suitable ranges of and p and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

AB - The aim of this paper is to study the following problem (P) ≡{ (-Δ)su = uq + up in ; u > 0 inΩ; Bsu = 0 in RN/Ω with 0 < q < 1 < p, N > 2s, > 0,Ω⊂ RN is a smooth bounded domain, (-Δ)su(x) = aN;s P:V: Z RN u(x) -u(y)/x-y/N+2s dy; aN;s is a normalizing constant, and Bsu = uX∑1 + NsuX∑2 : Here, ∑1 and ∑2 are open sets in RN/Ω such that∑2 ∩ ∑2 = ; and ∑1 ∩ ∑2 = RN/Ω: In this setting, Nsu can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [20], and Bsu is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem (P) for suitable ranges of and p and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

KW - Fractional laplacian

KW - Integro differential operators

KW - Mixed boundary condition

KW - Multiplicity of positive solution

KW - Weak solutions

UR - http://www.scopus.com/inward/record.url?scp=85044415238&partnerID=8YFLogxK

U2 - 10.3934/cpaa.2018053

DO - 10.3934/cpaa.2018053

M3 - Article

VL - 17

SP - 1103

EP - 1120

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

SN - 1534-0392

IS - 3

ER -