A semilinear elliptic equation with competing powers and a radial potential

2017/12/12 by Monica Musso, M. Musso, Musso, M. +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Combinatorics #FOS: Mathematics #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Partial Differential Equations #Physics #Position (finance) #math.AP

paper · pdf · doi:10.48550/arxiv.1712.04293

published in arXiv (Cornell University) (Cornell University) · 19 pages

arxiv created 2017/12/12 · openalex publication_date 2017/12/12 · arxiv updated 2017/12/13 · openalex created_date 2017/12/22 · openalex updated_date 2026/08/04

Abstract

We verify the existence of radial positive solutions for the semi-linear equation - Δu=u^p - V(y) u^q, u>0, in ℝ^N where N≥ 3, p is close to p^*:=(N+2)/(N-2), and V is a radial smooth potential. If q is super-critical, namely q>p^*, we prove that this Problem has a radial solution behaving like a super-position of bubbles blowing-up at the origin with different rates of concentration, provided V(0)<0. On the other hand, if N/(N-2)<q<p^*, we prove that this Problem has a radial solution behaving like a super-position of \it flat bubbles with different rates of concentration, provided lim_{r → ∞} V(r) <0.

Citations

• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

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