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(θ+\int_Ω |\nabla u| dx\right)Δu =u^p \quad{\rm in}\quad Ω\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial Ω, \end{equation*} where $p>1$, $\theta \in \mathbb{R}$, $\Omega$ is a bounded smooth domain containing the origin in ${\mathbb{R}}^N$ with $N\ge 2$. In the subcritical case: $1<p<N/(N-2)$ if $N\ge 3$, $1<p<+\infty$ 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 $$-\Delta u+\lambda u^p=k{\delta }_0\mathrm{in}{B}_1(0),u=0\mathrm{on}\partial B_1(0).$$ In the supercritical case: $N/(N-2)\le p<(N+2)/(N-2)$ 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.

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