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 -