Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908944548495360 |
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| author | Zhang, Yibin |
| author_facet | Zhang, Yibin |
| contents | Very differently from those perturbative techniques of Deng-Musso in [26], 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 $-Δu=λu|u|^{p-2}e^{|u|^p}$ under homogeneous Dirichlet boundary condition in a bounded, smooth planar domain $Ω$, when $0<p<2$ and $λ>0$ 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 $4πm$ from below, but for any $1<p<2$ the energy always converges to $4πm$ from above, where the latter case sharply recurs a result of De Marchis-Malchiodi-Martinazzi-Thizy in [32] involving concentration and compactness properties at any critical energy level $4πm$ 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 $λ$ small enough, we prove that when $Ω$ 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 $Ω$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$ Zhang, Yibin Analysis of PDEs Very differently from those perturbative techniques of Deng-Musso in [26], 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 $-Δu=λu|u|^{p-2}e^{|u|^p}$ under homogeneous Dirichlet boundary condition in a bounded, smooth planar domain $Ω$, when $0<p<2$ and $λ>0$ 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 $4πm$ from below, but for any $1<p<2$ the energy always converges to $4πm$ from above, where the latter case sharply recurs a result of De Marchis-Malchiodi-Martinazzi-Thizy in [32] involving concentration and compactness properties at any critical energy level $4πm$ 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 $λ$ small enough, we prove that when $Ω$ 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 $Ω$ 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. |
| title | Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.07641 |