The Loewner-Nirenberg problem in singular domains (Q2186610)

In this paper the asymptotic behaviors of solutions of the Loewner-Nirenberg problem \[ \Delta u=\frac{n(n-2)}{4u^{\frac{n+2}{n-2}}} \text{ in } \Omega,\quad u=\infty \text{ on } \partial\Omega,\tag{*} \] where \(\Omega\subset\mathbb R^n\) and \(n\ge3\), near singular points on \(\partial\Omega\), not necessarily isolated, are studied. It is proved that the solutions of (*) can be approximated by the corresponding solutions in tangent cones at singular points on the boundary, under appropriate conditions of the domain near singular points. The main result is: Theorem 1.1. Let \(\Omega\subset\mathbb R^n\) be a bounded Lipschitz domain with \(x_0\in\partial\Omega\) and, for some integer \(k\le n\), let \(\partial\Omega\) in a neighborhood of \(x_0\) consist of \(k\) \(C^{1,1}\)-hypersurfaces \(S_1,\dots,S_k\) intersecting at \(x_0\) with the property that the normal vectors of \(S_1,\dots,S_k\) at \(x_0\) are linearly independent. Suppose \(u\in C^{\infty}(\Omega)\) is a solution of (*), and \(u_{V_{x_0}}\) is the corresponding solution in the tangent cone \(V_{x_0}\) of \(\Omega\) at \(x_0\). Then, there exist a constant \(r\) and a \(C^{1,1}\)-diffeomorphism \(T: B_r(x_0)\to T(B_r(x_0))\subset\mathbb R^n\), with \(T(\Omega\cap B_r(x_0)) =V_{x_0} \cap T(B_r(x_0))\) and \(T(\partial\Omega\cap B_r(x_0)) =\partial V_{x_0} \cap T(B_r(x_0))\), such that, for any \(x\in B_{r/2}(x_0)\), \(\big|\dfrac{u(x)}{u_{V_{x_0}}(Tx)}-1\big|\le C|x - x_0|\), where \(C\) is a positive constant depending only on \(n\) and the geometry of \(\partial\Omega\).