Mass concentration of minimizers for $L^2$-subcritical Kirchhoff energy functional in bounded domains
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2025
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| author | Yang, Chen Yu, Shubin Tang, Chun-Lei |
| author_facet | Yang, Chen Yu, Shubin Tang, Chun-Lei |
| contents | We are concerned with $L^2$-constraint minimizers for the Kirchhoff functional $$ E_b(u)=\int_Ω|\nabla u|^2\mathrm{d}x+\frac{b}{2}\left(\int_Ω|\nabla u|^2\mathrm{d}x\right)^2+\int_ΩV(x)u^2\mathrm{d}x-\fracβ{2}\int_Ω|u|^4\mathrm{d}x, $$ where $b>0$, $β>0$ and $V(x)$ is a trapping potential in a bounded domain $Ω$ of $\mathbb R^2$. As is well known that minimizers exist for any $b>0$ and $β>0$, while the minimizers do not exist for $b=0$ and $β\geqβ^*$, where $β^*=\int_{\mathbb R^2}|Q|^2\mathrm{d}x$ and $Q$ is the unique positive solution of $-Δu+u-u^3=0$ in $\mathbb R^2$. In this paper, we show that for $β=β^*$, the energy converges to 0, but for $β>β^*$, the minimal energy will diverge to $-\infty$ as $b\searrow0$. Further, we give the refined limit behaviors and energy estimates of minimizers as $b\searrow0$ for $β=β^*$ or $β>β^*$. For both cases, we obtain that the mass of minimizers concentrates either at an inner point or near the boundary of $Ω$, depending on whether $V(x)$ attains its flattest global minimum at an inner point of $Ω$ or not. Meanwhile, we find an interesting phenomenon that the blow-up rate when the minimizers concentrate near the boundary of $Ω$ is faster than concentration at an interior point if $β=β^*$, but the blow-up rates remain consistent if $β>β^*$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_20300 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mass concentration of minimizers for $L^2$-subcritical Kirchhoff energy functional in bounded domains Yang, Chen Yu, Shubin Tang, Chun-Lei Analysis of PDEs 35J60, 35B40 We are concerned with $L^2$-constraint minimizers for the Kirchhoff functional $$ E_b(u)=\int_Ω|\nabla u|^2\mathrm{d}x+\frac{b}{2}\left(\int_Ω|\nabla u|^2\mathrm{d}x\right)^2+\int_ΩV(x)u^2\mathrm{d}x-\fracβ{2}\int_Ω|u|^4\mathrm{d}x, $$ where $b>0$, $β>0$ and $V(x)$ is a trapping potential in a bounded domain $Ω$ of $\mathbb R^2$. As is well known that minimizers exist for any $b>0$ and $β>0$, while the minimizers do not exist for $b=0$ and $β\geqβ^*$, where $β^*=\int_{\mathbb R^2}|Q|^2\mathrm{d}x$ and $Q$ is the unique positive solution of $-Δu+u-u^3=0$ in $\mathbb R^2$. In this paper, we show that for $β=β^*$, the energy converges to 0, but for $β>β^*$, the minimal energy will diverge to $-\infty$ as $b\searrow0$. Further, we give the refined limit behaviors and energy estimates of minimizers as $b\searrow0$ for $β=β^*$ or $β>β^*$. For both cases, we obtain that the mass of minimizers concentrates either at an inner point or near the boundary of $Ω$, depending on whether $V(x)$ attains its flattest global minimum at an inner point of $Ω$ or not. Meanwhile, we find an interesting phenomenon that the blow-up rate when the minimizers concentrate near the boundary of $Ω$ is faster than concentration at an interior point if $β=β^*$, but the blow-up rates remain consistent if $β>β^*$. |
| title | Mass concentration of minimizers for $L^2$-subcritical Kirchhoff energy functional in bounded domains |
| topic | Analysis of PDEs 35J60, 35B40 |
| url | https://arxiv.org/abs/2503.20300 |