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

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 $D_2=\frac{2N+8-Np}{4N(p-2)}$.