Existence and nonexistence of a positive solution of the Lane-Emden equation having a boundary singularity: the subcritical case (Q1662403)

The paper studies the problem \(-\Delta u=|u|^{p-1}u\) in \(\Omega \), \(u=\lambda \delta_x \) on \(\partial \Omega \). Here \(\Omega \) is a bounded uniform domain in \(\mathbb R^m\) with \(m\geq 3\) and \(x\in \partial \Omega \). It is shown that there exist \(p_0\in (1,\infty )\) and \(\lambda_0>0\) such that, if \(p\in (1,p_0)\) and \(\lambda >0\), then there exists a positive classical solution of the problem if and only if \(\lambda \leq \lambda_0\).