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In this paper, we study isolated singular positive solutions for thefollowing Kirchhoff--type Laplacian problem: \begin{equation*}-\left(\theta+\int_{\Omega} |\nabla u| dx\right)\Delta u =u^p \quad{\rmin}\quad \Omega\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial \Omega,\end{equation*} where $p>1$, $\theta\in \R$, $\Omega$ is a bounded smoothdomain containing the origin in $\R^N$ with $N\ge 2$. In the subcritical case:$1<p<N/(N-2)$ if $N\ge3$, $1<p<+\infty$ if $N=2$, we employ the Schauderfixed-point theorem to derive a sequence of positive isolated singularsolutions for the above problem such that $M_\theta(u)>0$. To estimate$M_\theta(u)$, we make use of the rearrangement argument. Furthermore, weobtain a sequence of isolated singular solutions such that $M_\theta(u)<0$, byanalyzing relationship between the parameter $\lambda$ and the unique solution$u_\lambda$ of $$-\Delta u+\lambda u^p=k\delta_0\quad{\rm in}\quadB_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0).$$ In the supercriticalcase: $N/(N-2)\le p<(N+2)/(N-2)$ with $N\ge3$, we obtain two isolated singularsolutions $u_i$ with $i=1,2$ such that $M_\theta(u_i)>0$ under some appropriateassumptions.

On isolated singularities of Kirchhoff--type Laplacian problems