Normalized solutions of $L^2$-supercritical Kirchhoff equations in bounded domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914929652531200 |
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| author | Wang, Qun Chang, Xiaojun |
| author_facet | Wang, Qun Chang, Xiaojun |
| contents | In this paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_Ω\vert\nabla u\vert^2dx)Δu+λu=\vert u\vert^{p-2}u & \text{ in }Ω,\\ u=0 & \text{ on }\partialΩ\end{cases} \end{equation*} subject to the constraint $\int_Ω\vert u\vert^2dx=c$. Here, $a$ and $b$ are positive constants, $Ω$ is a smooth bounded domain in $\mathbb{R}^N$ with $1\leq N\leq3$, $c>0$ is a prescribed value, and $λ\in \mathbb{R}$ is a Lagrange multiplier. In the $L^2$-supercritical regime $2+\frac{8}{N}<p<2^*$, we establish the existence of mountain pass-type normalized solutions. Our approach relies on utilizing a parameterized version of the minimax theorem with Morse index information for constraint functionals, and developing a blow-up analysis for the nonlinear Kirchhoff equations. Furthermore, we explore the asymptotic behavior of these solutions as $b\rightarrow0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_17155 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solutions of $L^2$-supercritical Kirchhoff equations in bounded domains Wang, Qun Chang, Xiaojun Analysis of PDEs 35J60, 35B09, 35B40, 47J30 In this paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_Ω\vert\nabla u\vert^2dx)Δu+λu=\vert u\vert^{p-2}u & \text{ in }Ω,\\ u=0 & \text{ on }\partialΩ\end{cases} \end{equation*} subject to the constraint $\int_Ω\vert u\vert^2dx=c$. Here, $a$ and $b$ are positive constants, $Ω$ is a smooth bounded domain in $\mathbb{R}^N$ with $1\leq N\leq3$, $c>0$ is a prescribed value, and $λ\in \mathbb{R}$ is a Lagrange multiplier. In the $L^2$-supercritical regime $2+\frac{8}{N}<p<2^*$, we establish the existence of mountain pass-type normalized solutions. Our approach relies on utilizing a parameterized version of the minimax theorem with Morse index information for constraint functionals, and developing a blow-up analysis for the nonlinear Kirchhoff equations. Furthermore, we explore the asymptotic behavior of these solutions as $b\rightarrow0$. |
| title | Normalized solutions of $L^2$-supercritical Kirchhoff equations in bounded domains |
| topic | Analysis of PDEs 35J60, 35B09, 35B40, 47J30 |
| url | https://arxiv.org/abs/2408.17155 |