On isolated singularities of Kirchhoff--type Laplacian problems

In this paper, we study isolated singular positive solutions for the following Kirchhoff–type Laplacian problem: \begin{equation*} -\left(\theta+\int_{\Omega} |\nabla u| dx\right)\Delta u =u^p \quad{\rm in}\quad \Omega\setminus {0},\qquad u=0\quad {\rm on}\quad \partial \Omega, \end{equation*} where $p>1$, $\theta \in \mathbb{R}$, $\mathrm{\Omega }$ is a bounded smooth domain containing the origin in ${\mathbb{R}}^N$ with $N\ge 2$. In the subcritical case: $1<p<N\mathrm{/}(N-2)$ if $N\ge 3$, $1<p<+\mathrm{\infty }$ if $N=2$, we employ the Schauder fixed-point theorem to derive a sequence of positive isolated singular solutions for the above problem such that $M_{\theta }(u)>0$. To estimate $M_{\theta }(u)$, we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that $M_{\theta }(u)<0$, by analyzing relationship between the parameter $\lambda$ and the unique solution $u_{\lambda }$ of $$-\mathrm{\Delta }u+\lambda u^p=k{\delta }_0\mathrm{i}\mathrm{n}{B}_1(0),u=0\mathrm{o}\mathrm{n}\mathrm{\partial }{B}_1(0)\mathrm{.}$$ In the supercritical case: $N\mathrm{/}(N-2)\le p<(N+2)\mathrm{/}(N-2)$ with $N\ge 3$, we obtain two isolated singular solutions $u_i$ with $i=1,2$ such that $M_{\theta }(u_i)>0$ under some appropriate assumptions.