In this paper, we study that the nearly critical nonlocal problem − ∆u = (|x| −(n−2) ∗ u p−ϵ )u p−1−ϵ in Ω, u > 0 in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R n for n = 3, 4, 5, ∗ denotes the standard convolution, ϵ > 0 is a small parameter and p = n+2 n−2 is energy-critical exponent. We study the asymptotic behavior of least energy solutions as ϵ → 0. These solutions are shown to blow-up at exactly one point x0 and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied. Finally, in order to further locate the blowing-up point x0, we prove that x0 is a global maximum point of the Robin’s function of Ω
Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent
SILVIA CINGOLANI
;SHUNNENG ZHAO
2026-01-01
Abstract
In this paper, we study that the nearly critical nonlocal problem − ∆u = (|x| −(n−2) ∗ u p−ϵ )u p−1−ϵ in Ω, u > 0 in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R n for n = 3, 4, 5, ∗ denotes the standard convolution, ϵ > 0 is a small parameter and p = n+2 n−2 is energy-critical exponent. We study the asymptotic behavior of least energy solutions as ϵ → 0. These solutions are shown to blow-up at exactly one point x0 and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied. Finally, in order to further locate the blowing-up point x0, we prove that x0 is a global maximum point of the Robin’s function of ΩI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
