English

Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms

Analysis of PDEs 2023-01-13 v1

Abstract

We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term {Δuμux2=u22εuin Ω, u=0on Ω, \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\varepsilon}u &&\quad \text{in } \Omega, \\\ u &= 0&&\quad \text{on } \partial\Omega, \end{aligned} \right. in a bounded domain ΩRN(N7)\Omega\subset\mathbb{R}^N (N\ge7) with 0Ω0\in\Omega, as μ,ε0+\mu,\varepsilon\to 0^+. In \cite{BarGuo-ANS}, we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as μ=O(ϵα)\mu=O(\epsilon^\alpha) with α>N4N2\alpha>\frac{N-4}{N-2}, ε0+\varepsilon\to 0^+. Here we prove the existence of nodal bubble tower solutions, i.e.\ superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order, as μ=O(ε)\mu=O(\varepsilon), ε0+\varepsilon\to0^+.

Keywords

Cite

@article{arxiv.2301.04928,
  title  = {Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms},
  author = {Thomas Bartsch and Qianqiao Guo},
  journal= {arXiv preprint arXiv:2301.04928},
  year   = {2023}
}

Comments

18 pages. arXiv admin note: text overlap with arXiv:1508.04007

R2 v1 2026-06-28T08:10:07.060Z