On the concentration phenomenon of $L^2$-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

TL;DR

This paper investigates the existence and concentration behavior of minimizers for a class of Kirchhoff equations with potentials, revealing their asymptotic form and conditions for existence in subcritical regimes.

Contribution

It provides the first detailed analysis of the concentration phenomena of constrained minimizers for Kirchhoff equations with potentials, including explicit asymptotic profiles.

Findings

Existence of global minimizers under certain potential conditions.

Asymptotic profile of minimizers for large constraint c.

Explicit formulas for the concentration points and profiles.

Abstract

In this paper, we study the existence and the concentration behavior of minimizers for $i_V(c)=\inf\limits_{u\in S_c}I_V(u)$, here $S_c=\{u\in H^1(\R^N)|~\int_{\R^N}V(x)|u|^2<+\infty,~|u|_2=c>0\}$ and $$I_V(u)=\frac{1}{2}\ds\int_{\R^N}(a|\nabla u|^2+V(x)|u|^2)+\frac{b}{4}\left(\ds\int_{\R^N}|\nabla u|^2\right)^2-\frac{1}{p}\ds\int_{\R^N}|u|^{p},$$ where $N=1,2,3$ and $a,b>0$ are constants. By the Gagliardo-Nirenberg inequality, we get the sharp existence of global constraint minimizers for $2<p<2^*$ when $V(x)\geq0$, $V(x)\in L^{\infty}_{loc}(\R^N)$ and $\lim\limits_{|x|\rightarrow+\infty}V(x)=+\infty$. For the case $p\in(2,\frac{2N+8}{N})\backslash\{4\}$, we prove the global constraint minimizers $u_c$ behave like $$ u_{c}(x)\approx \frac{c}{|Q_{p}|_2}\left(\frac{m_{c}}{c}\right)^{\frac{N}{2}}Q_p\left(\frac{m_{c}}{c}x-z_c\right).$$ for some $z_c\in\R^N$ when $c$ is large, where $Q_p$ is up to translations, the unique positive solution of $-\frac{N(p-2)}{4}\Delta Q_p+\frac{2N-p(N-2)}{4}Q_p=|Q_p|^{p-2}Q_p$ in $\R^N$ and $m_c=(\frac{\sqrt{a^2D_1^2-4bD_2i_0(c)}+aD_1}{2bD_2})^{\frac12}$, $D_1=\frac{Np-2N-4}{2N(p-2)}$ and $D_2=\frac{2N+8-Np}{4N(p-2)}$.