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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.16634 |
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| _version_ | 1866911522901458944 |
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| author | Tang, Zhi-Yun Li, Gui-Dong Li, Yong-Yong |
| author_facet | Tang, Zhi-Yun Li, Gui-Dong Li, Yong-Yong |
| contents | This paper focuses on the critical Kirchhoff equation with concave perturbation
\begin{align*}
\begin{cases}
\displaystyle -\Big(a+b\int_Ω|\nabla u|^2dx\Big)Δu=|u|^4u+λ|u|^{q-2}u\ \ &\mbox{in}\ Ω,
\displaystyle u=0\ \ &\mbox{on}\ \partialΩ,
\end{cases}
\end{align*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^3$, $a,b,λ>0$ and $1<q<2$. By the constrained minimization methods, the mountain pass theorem and the concentration-compactness principle, we verify the multiplicity of positive solutions for $λ>0$ small enough. Moreover, we analyse the asymptotic behaviour of positive solutions as $b\rightarrow0$ and $λ\rightarrow0$, respectively. This work is a counterpart of [A. Ambrosetti et al., J.~Funct.~Anal. 1994] for the Kirchhoff equation. It is noteworthy that we don't require that $b>0$ is small enough here, which is imposed in the existing literatures to make refined estimates for the mountain pass level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16634 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplicity and asymptotics of positive solutions for critical-concave Kirchhoff equation Tang, Zhi-Yun Li, Gui-Dong Li, Yong-Yong Analysis of PDEs 35J20, 35B33, 35B38, 35D30 This paper focuses on the critical Kirchhoff equation with concave perturbation \begin{align*} \begin{cases} \displaystyle -\Big(a+b\int_Ω|\nabla u|^2dx\Big)Δu=|u|^4u+λ|u|^{q-2}u\ \ &\mbox{in}\ Ω, \displaystyle u=0\ \ &\mbox{on}\ \partialΩ, \end{cases} \end{align*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^3$, $a,b,λ>0$ and $1<q<2$. By the constrained minimization methods, the mountain pass theorem and the concentration-compactness principle, we verify the multiplicity of positive solutions for $λ>0$ small enough. Moreover, we analyse the asymptotic behaviour of positive solutions as $b\rightarrow0$ and $λ\rightarrow0$, respectively. This work is a counterpart of [A. Ambrosetti et al., J.~Funct.~Anal. 1994] for the Kirchhoff equation. It is noteworthy that we don't require that $b>0$ is small enough here, which is imposed in the existing literatures to make refined estimates for the mountain pass level. |
| title | Multiplicity and asymptotics of positive solutions for critical-concave Kirchhoff equation |
| topic | Analysis of PDEs 35J20, 35B33, 35B38, 35D30 |
| url | https://arxiv.org/abs/2603.16634 |