On isolated singularities of Kirchhoff--type Laplacian problems

Abstract

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 $p>1$, $\theta\in \R$, $\Omega$ is a bounded smooth domain 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 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 $$-\Delta u+\lambda u^p=k\delta_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0).$$ In the supercritical case: $N/(N-2)\le p<(N+2)/(N-2)$ with $N\ge3$, we obtain two isolated singular solutions $u_i$ with $i=1,2$ such that $M_\theta(u_i)>0$ under some appropriate assumptions.