Asymptotic behaviour of the least energy solutions of fractional semilinear Neumann problem

2023/01/09 by Somnath Gandal, Gandal, Somnath, Jagmohan Tyagi +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.03260

openalex publication_date 2023/01/09 · openalex created_date 2023/01/11 · openalex updated_date 2026/07/28

Abstract

We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem: \ d(-Δ)^su+ u= \absu^p-1u in Ω, N_su=0 \text in ℝⁿ∖ Ω, u · gt;0 in Ω, . where Ω⊂ ℝⁿ is a bounded domain of class C^1,1, 1 max \1, 2s \, 00 and N_su is the nonlocal Neumann derivative. We show that for small d, the least energy solutions u_d of the above problem achieves L^∞ bound independent of d. Using this together with suitable L^r-estimates on u_d, we show that least energy solution u_d achieve maximum on the boundary of Ω for d sufficiently small.

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