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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.11037 |
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| _version_ | 1866916524638339072 |
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| author | Zhou, Jianwen Yang, Puming |
| author_facet | Zhou, Jianwen Yang, Puming |
| contents | In this paper, we study a type of p-Kirchhoff equation $$
-\left( a+b\int_{\mathbb{R} ^3}{\left| \nabla u \right|^pdx} \right) \varDelta _pu=λ\left| u \right|^{p-2}u+\left| u \right|^{q-2}u, x \in \mathbb{R}^3 $$ with the prescribed mass $$ \left(\int_{\mathbb{R} ^3}{\left| u \right|^{p}dx}\right)^\frac{1}{p} = c > 0 $$ where $a>0, b > 0,\frac{3}{2} <p <3, p < q < p^{\ast}:=\frac{3p}{3-p} $,$\varDelta _pu=div\left( \left| \nabla u \right|^{p-2}\nabla u \right)$ is the $p$-Laplacian of $u$, $λ\in \mathbb{R}$ is Lagrange multiplier. We consider both $L^p$-subcritical , $L^p$-critical and $L^p$-supercritical cases. Precisely, in the $L^p$-subcritical and $L^p$-critical cases, we obtain the existence and nonexistence of the normalized solutions for the $p$-Kirchhoff equation. In the $L^p$-supercritical case, we obtain the existence of radial ground sates and multiplicity of radial normalized solutions for the $p$-Kirchhoff equation. Furthermore, we study the asymptotic behavior of normalized solutions when $b \rightarrow 0^+$. Besides, when $\frac{3}{2} < p \leq 2$, benefit from the uniqueness(up to translations) of optimizer for Gargliardo-Nirenberg inequality, we show the existence and uniqueness of normalized solutions and provide the accurate descriptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11037 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplicity and asymptotic behavior of normalized solutions to p-Kirchhoff equations Zhou, Jianwen Yang, Puming Analysis of PDEs In this paper, we study a type of p-Kirchhoff equation $$ -\left( a+b\int_{\mathbb{R} ^3}{\left| \nabla u \right|^pdx} \right) \varDelta _pu=λ\left| u \right|^{p-2}u+\left| u \right|^{q-2}u, x \in \mathbb{R}^3 $$ with the prescribed mass $$ \left(\int_{\mathbb{R} ^3}{\left| u \right|^{p}dx}\right)^\frac{1}{p} = c > 0 $$ where $a>0, b > 0,\frac{3}{2} <p <3, p < q < p^{\ast}:=\frac{3p}{3-p} $,$\varDelta _pu=div\left( \left| \nabla u \right|^{p-2}\nabla u \right)$ is the $p$-Laplacian of $u$, $λ\in \mathbb{R}$ is Lagrange multiplier. We consider both $L^p$-subcritical , $L^p$-critical and $L^p$-supercritical cases. Precisely, in the $L^p$-subcritical and $L^p$-critical cases, we obtain the existence and nonexistence of the normalized solutions for the $p$-Kirchhoff equation. In the $L^p$-supercritical case, we obtain the existence of radial ground sates and multiplicity of radial normalized solutions for the $p$-Kirchhoff equation. Furthermore, we study the asymptotic behavior of normalized solutions when $b \rightarrow 0^+$. Besides, when $\frac{3}{2} < p \leq 2$, benefit from the uniqueness(up to translations) of optimizer for Gargliardo-Nirenberg inequality, we show the existence and uniqueness of normalized solutions and provide the accurate descriptions. |
| title | Multiplicity and asymptotic behavior of normalized solutions to p-Kirchhoff equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.11037 |