Mass concentration of minimizers for $L^2$-subcritical Kirchhoff energy functional in bounded domains

TL;DR

This work analyzes L^2-constrained minimizers for the nonlocal Kirchhoff energy in a bounded 2D domain, focusing on the asymptotic regime b→0 under two nonlinear strengths β=β^* (L^2-subcritical but critical) and β>β^*. The authors establish refined blow-up profiles and energy asymptotics by comparing to an auxiliary problem on ℝ^2 and exploiting scales tied to the potential V’s flattest minima. They show that mass concentration occurs at inner flattest minima when such points exist, or near the boundary otherwise, with distinct blow-up rates: interior concentration yields simpler power-law scaling, while boundary concentration introduces logarithmic corrections (for β=β^*), and the blow-up rate is uniform across interior and boundary for β>β^*. These results advance the understanding of concentration phenomena in nonlocal Kirchhoff problems and provide precise asymptotics that link the geometry of V to the location and shape of minimizers in bounded domains.

Abstract

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 $β>β^*$.

Theorems & Definitions (35)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Theorem 1.7
• Theorem 1.9
• Lemma 3.1
• proof
• Lemma 3.2
• proof