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.