On nodal solutions with a prescribed number of nodes for a Kirchhoff-type problem
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915116024332288 |
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| author | Fan, Haining Squassina, Marco Zhang, Jianjun |
| author_facet | Fan, Haining Squassina, Marco Zhang, Jianjun |
| contents | We are concerned with the existence and asymptotic behavior of multiple radial sign-changing solutions with the nodal characterization for a Kirchhoff-type problem involving the nonlinearity $|u|^{p-2}u(2<p<4)$ in $\mathbb{R}^3$. By developing some useful analysis techniques and introducing a novel definition of the Nehari manifold for the auxiliary system of the equations, we show that, for any positive integer $k$, the problem has a sign-changing solution $u_k^b$ changing signs exactly $k$ times. Furthermore, the energy of $u_k^b$ is strictly increasing in $k$, as well as some asymptotic behaviors of $u_k^b$ are obtained. Our result is a complement of [Deng Y, Peng S, Shuai W, {\it J. Funct. Anal.}, {\bf269}(2015), 3500-3527], where the case $2<p<4$ is left open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On nodal solutions with a prescribed number of nodes for a Kirchhoff-type problem Fan, Haining Squassina, Marco Zhang, Jianjun Analysis of PDEs 35A15, 35B09, 35B38, 35J20 We are concerned with the existence and asymptotic behavior of multiple radial sign-changing solutions with the nodal characterization for a Kirchhoff-type problem involving the nonlinearity $|u|^{p-2}u(2<p<4)$ in $\mathbb{R}^3$. By developing some useful analysis techniques and introducing a novel definition of the Nehari manifold for the auxiliary system of the equations, we show that, for any positive integer $k$, the problem has a sign-changing solution $u_k^b$ changing signs exactly $k$ times. Furthermore, the energy of $u_k^b$ is strictly increasing in $k$, as well as some asymptotic behaviors of $u_k^b$ are obtained. Our result is a complement of [Deng Y, Peng S, Shuai W, {\it J. Funct. Anal.}, {\bf269}(2015), 3500-3527], where the case $2<p<4$ is left open. |
| title | On nodal solutions with a prescribed number of nodes for a Kirchhoff-type problem |
| topic | Analysis of PDEs 35A15, 35B09, 35B38, 35J20 |
| url | https://arxiv.org/abs/2501.12865 |