A unified approach to symmetry for semilinear equations associated to the Laplacian in ℝ^N

2019/03/20 by Andrés I. Ávila, Avila, Andres I., Friedemann Brock +1
Computer Science · Mathematics · #26D10 #35J20 #35J60 #46E35 #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.1903.08626

openalex publication_date 2019/03/20 · openalex created_date 2019/04/01 · openalex updated_date 2026/07/28

Abstract

We show radial symmetry of positive solutions to the Hénon equation -Δu = |x|^-ℓ u^q in ℝ^N ∖ \ 0\ , where ℓ ≥ 0, q>0 and satisfy further technical conditions. A new ingredient is a maximum principle for open subsets of a half space. It allows to apply the Moving Plane Method once a slow decay of the solution at infinity has been established, that is lim _{|x|→ ∞} |x|^γ u(x) =L , for some numbers γ∈ (0, N-2) and L >0. Moreover, some examples of non-radial solutions are given for q> (N+1)/(N-3) and N≥ 4. We also establish radial symmetry for related and more general problems in ℝ^N and ℝ^N ∖ \ 0\ .

Citations

• A symmetry problem in potential theory
• Global and local behavior of positive solutions of nonlinear elliptic equations
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

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