A nonlocal concave-convex problem with nonlocal mixed boundary data

Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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
Issue number3
Publication statusPublished - 1 May 2018


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