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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2409.05130 |
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| _version_ | 1866918286889844736 |
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| author | Yu, Shubin Yang, Chen Tang, Chun-Lei |
| author_facet | Yu, Shubin Yang, Chen Tang, Chun-Lei |
| contents | In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation $$ \left\{\begin{array}{ll} -(a+b\int_Ω|\nabla u|^2\mathrm{d}x)Δu+V(x)u=μu+β^*|u|^{\frac{8}{3}}u &\mbox{in}\ Ω, \\[0.1cm]
u=0&\mbox{on}\ {\partialΩ}, \\[0.1cm] \int_Ω|u|^2\mathrm{d}x=1, \\[0.1cm] \end{array} \right. $$ where $a\geq0$, $b>0$, the function $V(x)$ is a trapping potential in a bounded domain $Ω\subset\mathbb R^3$, $β^*:=\frac{b}{2}|Q|_2^{\frac{8}{3}}$ and $Q$ is the unique positive radially symmetric solution of equation $-2Δu+\frac{1}{3}u-|u|^{\frac{8}{3}}u=0.$ We consider the existence of constraint minimizers for the associated energy functional involving the parameter $a$. The minimizer corresponds to the normalized ground state of above problem, and it exists if and only if $a>0$. Moreover, when $V(x)$ attains its flattest global minimum at an inner point or only at the boundary of $Ω$, we analyze the fine limit profiles of the minimizers as $a\searrow 0$, including mass concentration at an inner point or near the boundary of $Ω$. In particular, we further establish the local uniqueness of the minimizer if it is concentrated at a unique inner point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concentration behavior of normalized ground states for mass critical Kirchhoff equations in bounded domains Yu, Shubin Yang, Chen Tang, Chun-Lei Analysis of PDEs In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation $$ \left\{\begin{array}{ll} -(a+b\int_Ω|\nabla u|^2\mathrm{d}x)Δu+V(x)u=μu+β^*|u|^{\frac{8}{3}}u &\mbox{in}\ Ω, \\[0.1cm] u=0&\mbox{on}\ {\partialΩ}, \\[0.1cm] \int_Ω|u|^2\mathrm{d}x=1, \\[0.1cm] \end{array} \right. $$ where $a\geq0$, $b>0$, the function $V(x)$ is a trapping potential in a bounded domain $Ω\subset\mathbb R^3$, $β^*:=\frac{b}{2}|Q|_2^{\frac{8}{3}}$ and $Q$ is the unique positive radially symmetric solution of equation $-2Δu+\frac{1}{3}u-|u|^{\frac{8}{3}}u=0.$ We consider the existence of constraint minimizers for the associated energy functional involving the parameter $a$. The minimizer corresponds to the normalized ground state of above problem, and it exists if and only if $a>0$. Moreover, when $V(x)$ attains its flattest global minimum at an inner point or only at the boundary of $Ω$, we analyze the fine limit profiles of the minimizers as $a\searrow 0$, including mass concentration at an inner point or near the boundary of $Ω$. In particular, we further establish the local uniqueness of the minimizer if it is concentrated at a unique inner point. |
| title | Concentration behavior of normalized ground states for mass critical Kirchhoff equations in bounded domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.05130 |