Multi-bubble nodal solutions to slightly subcritical elliptic problems with Hardy terms

2015/08/17 by Thomas Bartsch, Bartsch, Thomas, Qianqiao Guo +1
Computer Science · Mathematics · #35J20 #35J60 #35J70 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Primary: 35B44 #Secondary: 35B33

paper · pdf · doi:10.48550/arxiv.1508.04007

openalex publication_date 2015/08/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

The paper is concerned with the slightly subcritical elliptic problem with Hardy term \ \beginaligned -Δu-μ(u)/(|x|²) amp;= |u|^2^∗-2-εu amp;amp; in Ω,
u amp;= 0amp;amp; on ∂Ω, \endaligned . in a bounded domain Ω⊂ℝ^N with 0∈Ω, in dimensions N≥7. We prove the existence of multi-bubble nodal solutions that blow up positively at the origin and negatively at a different point as ε→0 and μ=ε^α with α>(N-4)/(N-2). In the case of Ω being a ball centered at the origin we can obtain solutions with up to 5 bubbles of different signs. We also obtain nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order. The asymptotic shape of the solutions is determined in detail.

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