Ground state solution of a Kirchhoff type equation with singular potentials

Abstract

We study the existence and blow-up behavior of minimizers for $E(b)=\inf \{{\mathcal{E}}_b(u)|u\in H^1(R^2),\Vert u{\Vert }_{L^2}=1\},$ here ${\mathcal{E}}_b(u)$ is the Kirchhoff energy functional defined by ${\mathcal{E}}_b(u)={\int }_{R^2}|\nabla u{|}^{2}dx+b({\int }_{R^2}|\nabla u{|}^{2}dx{)}^2+{\int }_{R^2}V(x)|u(x){|}^{2}dx-\frac{a}{2}{\int }_{R^2}|u{|}^{4}dx,$ where $a>0$ and $b>0$ are constants. When $V(x)=-|x{|}^{-p}$ with $0<p<2$, we prove that the problem has (at least) a minimizer that is non-negative and radially symmetric decreasing. For $a\ge a^{\ast }$ (where $a^{\ast }$ is the optimal constant in the Gagliardo-Nirenberg inequality), we get the behavior of $E(b)$ when $b\to 0^{+}$. Moreover, for the case $a=a^{\ast }$, we analyze the details of the behavior of the minimizers $u_b$ when $b\to 0^{+}$.

16 pages