论文标题
具有单数电位的基尔chhoff型方程的基态解决方案
Ground state solution of a Kirchhoff type equation with singular potentials
论文作者
论文摘要
我们研究了最小化器的存在和爆炸行为，以$ e（b）= \ inf \ big \ {\ Mathcal {e} _b（u）\，| \，| \，u \，u \ in H^1（r^2），\ | U \ | u \ | _ _ _ {l^2} = 1} = 1 \ big \ \ \ \ big \ big \ bim $ Kirchhoff Energy功能由 $ \ MATHCAL { \ int_ {r^2} | u |^4 dx，$ 其中$ a> 0 $和$ b> 0 $是常数。当$ v（x）= - | x |^{ - p} $带有$ 0 <p <2 $，我们证明该问题具有（至少）最小化，而最小化是非阴性和径向对称减少的。对于$ a \ ge a^*$（其中$ a^*$是gagliardo-nirenberg不平等中的最佳常数），当$ b \ to 0^+$时，我们会获得$ e（b）$的行为。此外，对于情况$ a = a^*$，我们分析了$ b \ to 0^+$时，我们分析了最小化器$ u_b $的行为的详细信息。
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 $b\to 0^+$.
