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 $i_V(c)=\underset{u\in S_c}{\inf }{I}_V(u)$, here $S_c=\{u\in H^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}V(x)|u{|}^2<+\infty ,|u{|}_2=c>0\}$ 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 $2<p<2^{\ast }$ when $V(x)\ge 0$, $V(x)\in L_{loc}^{\infty }({\mathbb{R}}^N)$ and $\underset{|x|\to +\infty }{\lim }V(x)=+\infty$. 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 $z_c\in {\mathbb{R}}^N$ 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 ${\mathbb{R}}^N$ and $m_c=(\frac{\sqrt{a^2{D}_1^2-4b{D}_2{i}_0(c)}+a{D}_1}{2b{D}_2}{)}^{\frac{1}{2}}$, $D_1=\frac{Np-2N-4}{2N(p-2)}$ and $D_2=\frac{2N+8-Np}{4N(p-2)}$.