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Article Dans Une Revue Topological Methods in Nonlinear Analysis Année : 2019

A Global multiplicity result for a very singular critical nonlocal equation

Résumé

In this article, we show the global multiplicity result for the following nonlocal singular problem (P λ) : (−∆) s u = u −q + λu 2 * s −1 , u > 0 in Ω, u = 0 in R n \ Ω, where Ω is a bounded domain in R n with smooth boundary ∂Ω, n > 2s, s ∈ (0, 1), λ > 0, q > 0 satisfies q(2s − 1) < (2s + 1) and 2 * s = 2n n−2s. Employing the variational method, we show the existence of at least two distinct weak positive solutions for (P λ) in X 0 when λ ∈ (0, Λ) and no solution when λ > Λ, where Λ > 0 is appropriately chosen. We also prove a result of independent interest that any weak solution to (P λ) is in C α (R n) with α = α(s, q) ∈ (0, 1). The asymptotic behaviour of weak solutions reveals that this result is sharp.
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Dates et versions

hal-02161068 , version 1 (20-06-2019)

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Jacques Giacomoni, T Mukherjee, K. Sreenadh. A Global multiplicity result for a very singular critical nonlocal equation. Topological Methods in Nonlinear Analysis, In press, ⟨10.12775/TMNA.2019.049⟩. ⟨hal-02161068⟩
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