Ground state solution of a Kirchhoff type equation with singular potentials

TL;DR

This paper investigates the existence, symmetry, and blow-up behavior of minimizers for a Kirchhoff energy functional with singular potentials in two dimensions, revealing conditions for minimizer existence and detailed asymptotic behaviors.

Contribution

It establishes the existence of radially symmetric minimizers for singular potentials and analyzes their asymptotic behavior as parameters approach critical values.

Findings

Existence of non-negative, radially symmetric minimizers for certain potentials.

Behavior of the energy functional as the parameter b approaches zero.

Asymptotic analysis of minimizers at the critical constant a*.

Abstract

We study the existence and blow-up behavior of minimizers for $E(b)=\inf\Big\{\mathcal{E}_b(u) \,|\, u\in H^1(R^2), \|u\|_{L^2}=1\Big\},$ here $\mathcal{E}_b(u)$ is the Kirchhoff energy functional defined by $\mathcal{E}_b(u)= \int_{R^2} |\nabla u|^2 dx+ b(\int_{R^2} |\nabla u|^2d x)^2+\int_{R^2} V(x) |u(x)|^2 dx - \frac{a}{2} \int_{R^2} |u|^4 dx,$ 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 $a\ge a^*$ (where $a^*$ is the optimal constant in the Gagliardo-Nirenberg inequality), we get the behavior of $E(b)$ when $b\to 0^+$. Moreover, for the case $a=a^*$, we analyze the details of the behavior of the minimizers $u_b$ when $b\to 0^+$.