PreprintCC0arXiv
A Supercritical Problem in Dimension Two
Abstract
Let Ω⊂ℝ2 be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let 0<λ<λ1(Ω), where Λ1 (Ω) is the first eigenvalue for −Δ on Ω under Dirichlet boundary conditions. For every fixed integer k≥1 and all sufficiently small ε>0, we construct a positive solution of −Δu=λu eu2+ε in Ω, u=0 on ∂Ω, which blows up at the origin as a tower of k Liouville bubbles. The concentration scales are strongly separated. The proof is based on a Lyapunov--Schmidt reduction adapted to these different scales.
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BibTeX
@misc{pino2026supercritical,
title = {{A Supercritical Problem in Dimension Two}},
author = {Manuel del Pino and Ignacio Guerra and Monica Musso},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22413},
eprint = {2609.22413},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22413},
note = {CC0},
}
RIS
TY - UNPB
TI - A Supercritical Problem in Dimension Two
AU - Manuel del Pino
AU - Ignacio Guerra
AU - Monica Musso
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22413
UR - https://arxiv.org/abs/2609.22413
AB - Let Ω⊂ℝ^2 be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let 0<λ<λ_1(Ω), where Λ_1 (Ω) is the first eigenvalue for −Δ on Ω under Dirichlet boundary conditions. For every fixed integer k≥1 and all sufficiently small ε>0, we construct a positive solution of −Δu=λu e^u^2+ε in Ω, u=0 on ∂Ω, which blows up at the origin as a tower of k Liouville bubbles. The concentration scales are strongly separated. The proof is based on a Lyapunov--Schmidt reduction adapted to these different scales.
ER -