On the Kirchhoff equation with prescribed mass and general nonlinearities
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913180831186944 |
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| author | Zeng, Xiaoyu Zhang, Jianjun Zhang, Yimin Zhong, Xuexiu |
| author_facet | Zeng, Xiaoyu Zhang, Jianjun Zhang, Yimin Zhong, Xuexiu |
| contents | In the present paper, we apply a global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions $(λ_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the following Kirchhoff problem $$ -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)Δu+λu=g(u)~\hbox{in}~\mathbb{R}^N,\;N\geq 1 $$ satisfying the normalization constraint $
\displaystyle\int_{\mathbb{R}^N}u^2=c, $ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as well as the mass $c>0$. Due to the presence of the non-local term $b\int_{\mathbb{R}^N}|\nabla u|^2dx Δu$, such problems lack the mountain pass geometry in the higher dimension case $N\geq 5$. Our result seems to be the first attempt in this aspect. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_10293 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Kirchhoff equation with prescribed mass and general nonlinearities Zeng, Xiaoyu Zhang, Jianjun Zhang, Yimin Zhong, Xuexiu Analysis of PDEs 35A01, 35B09, 35B40, 35J60 In the present paper, we apply a global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions $(λ_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the following Kirchhoff problem $$ -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)Δu+λu=g(u)~\hbox{in}~\mathbb{R}^N,\;N\geq 1 $$ satisfying the normalization constraint $ \displaystyle\int_{\mathbb{R}^N}u^2=c, $ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as well as the mass $c>0$. Due to the presence of the non-local term $b\int_{\mathbb{R}^N}|\nabla u|^2dx Δu$, such problems lack the mountain pass geometry in the higher dimension case $N\geq 5$. Our result seems to be the first attempt in this aspect. |
| title | On the Kirchhoff equation with prescribed mass and general nonlinearities |
| topic | Analysis of PDEs 35A01, 35B09, 35B40, 35J60 |
| url | https://arxiv.org/abs/2112.10293 |