Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$

[2403.07641] Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$ Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2403.07641 (math) [Submitted on 12 Mar 2024 ( v1 ), last revised 2 Sep 2026 (this version, v11)] Title: Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$ Authors: Yibin Zhang View a PDF of the paper titled Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$, by Yibin Zhang View PDF HTML (experimental) Abstract: Very differently from those perturbative techniques of Deng-Musso in [27], we use the assumption of a $C^1$-stable critical point to construct positive or sign-changing solutions with arbitrary $m$ isolated bubbles to the boundary value problem $-\Delta u=\lambda u|u|^{p-2}e^{|u|^p}$ under homogeneous Dirichlet boundary condition in a bounded, smooth planar domain $\Omega$, when $0<p<2$ and $\lambda>0$ is a small but free parameter. We build a vanishing identity of first order and an identity of second order to prove that for any $0<p<1$ the delicate energy expansion of these bubbling solutions always converges to $4\pi m$ from below, but for any $1<p<2$ the energy always converges to $4\pi m$ from above, where the latter case sharply recurs a result of De Marchis-Malchiodi-Martinazzi-Thizy in [33] involving concentration and compactness properties at any critical energy level $4\pi m$ only for positive bubbling solutions. A sufficient condition on the intersection between the nodal line of these sign-changing solutions and the boundary of the domain is founded. Moreover, for $\lambda$ small enough, we prove that when $\Omega$ is an arbitrary bounded domain, this problem has not only at least two pairs of bubbling solutions which change sign exactly once and whose nodal lines intersect the boundary, but also a bubbling solution which changes sign exactly twice or three times; when $\Omega$ has an axial symmetry, this problem has a bubbling solution which alternately changes sign arbitrarily many times along the axis of symmetry through the domain. Subjects: Analysis of PDEs (math.AP) Cite as: arXiv:2403.07641 [math.AP] (or arXiv:2403.07641v11 [math.AP] for this version) https://doi.org/10.48550/arXiv.2403.07641 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yibin Zhang [ view email ] [v1] Tue, 12 Mar 2024 13:25:22 UTC (678 KB) [v2] Mon, 25 Nov 2024 11:10:41 UTC (714 KB) [v3] Wed, 26 Mar 2025 07:29:09 UTC (734 KB) [v4] Mon, 23 Mar 2026 00:49:35 UTC (783 KB) [v5] Wed, 25 Mar 2026 01:16:15 UTC (50 KB) [v6] Wed, 8 Apr 2026 03:24:20 UTC (50 KB) [v7] Fri, 12 Jun 2026 00:53:58 UTC (50 KB) [v8] Sun, 21 Jun 2026 03:43:55 UTC (50 KB) [v9] Mon, 20 Jul 2026 14:49:42 UTC (50 KB) [v10] Sun, 16 Aug 2026 11:37:41 UTC (50 KB) [v11] Wed, 2 Sep 2026 01:02:43 UTC (50 KB) Full-text links: Access Paper: View a PDF of the paper titled Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$, by Yibin Zhang View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2024-03 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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