A nonlocal concave-convex problem with nonlocal mixed boundary data

Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1103-1120
Number of pages18
JournalCommunications on Pure and Applied Analysis
Volume17
Issue number3
DOIs
Publication statusPublished - 1 May 2018

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Dirichlet
Concave-convex Nonlinearities
Normalizing Constant
Fractional Laplacian
Neumann Condition
Open set
Nonexistence
Bounded Domain
Multiplicity
Energy
Interaction
Range of data
Interpretation

Cite this

Abdellaoui, Boumediene ; Dieb, Abdelrazek ; Valdinoci, Enrico. / A nonlocal concave-convex problem with nonlocal mixed boundary data. In: Communications on Pure and Applied Analysis. 2018 ; Vol. 17, No. 3. pp. 1103-1120.
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abstract = "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.",
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A nonlocal concave-convex problem with nonlocal mixed boundary data. / Abdellaoui, Boumediene; Dieb, Abdelrazek; Valdinoci, Enrico.

In: Communications on Pure and Applied Analysis, Vol. 17, No. 3, 01.05.2018, p. 1103-1120.

Research output: Contribution to journalArticle

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