Qualitative properties of positive solutions for mixed integro-differential equations

2017/10/10 by Ying Wang, Wang, Ying, partricio felmer +1
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1710.03413

openalex publication_date 2017/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is concerned with the qualitative properties of the solutions of mixed integro-differential equation \ \arraycolsep=1pt (-Δ)_x^α u+(-Δ)_y u+u=f(u) \rm in \R^N×\R^M, u · gt;0 \rmin \R^N×\R^M, lim_{|(x,y)|→+∞}u(x,y)=0, . with N≥ 1, M≥ 1 and α∈ (0,1). We study decay and symmetry properties of the solutions to this equation. Difficulties arise due to the mixed character of the integro-differential operators. Here, a crucial role is played by a version of the Hopf's Lemma we prove in our setting. In studying the decay, we construct appropriate super and sub solutions and we use the moving planes method to prove the symmetry properties.

Citations

• Uniqueness of positive solutions of Δu−u+up=0 in Rn

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