Solutions of some nonlinear elliptic problems, which are singular on a submanifold
Let \((M,g)\) be a \(C^2\) complete connected Riemannian manifold of dimension \(N\geq 3\) and \(\Sigma\) a compact submanifold of \(M\) of dimension \(d\geq 0\) without boundary. First we prove a removability result and secondly, if \(M\) is compact, we construct solutions of \[ -\Delta_g u(\cdot) +h(.,u(\cdot)) =0\text{ in }M \setminus \Sigma \] which blow-up on \(\Sigma\). More precisely: if \(h(x,r)= | r|^{q-1}r\), \(q>1\) and \((N<d+2\) and \(1<q<(N-d)/(N-d-2))\) or \((N=d+2\) and \(q>1)\), then for all \(\gamma\in [0,\infty]\), there exists a unique function \(u_\gamma\in C^1(M \setminus \Sigma)\), solution of the above equation and satisfying \(\lim_{\text{dist} (x,\Sigma)\to 0} u_\gamma(x)/ \mu(x)= \gamma\) where \(\mu\) is the fundamental solution of the Laplacian in \(\mathbb{R}^{N-d}\). Moreover if \(\gamma= \infty\), then \[ \lim_{\text{dist} (x,\Sigma) \to 0} \text{dist} (x,\Sigma)^{2/(q-1)} u_\infty(x) =\biggl[ \bigl(2/(q-1)\bigr) \bigl( 2q/(q-1)-N +d\bigr) \biggr]^{1/(q-1)}. \] We have a similar result when \(h(x,r) =e^r\) and \(N=d+2\). Those solutions play a role in Riemannian geometry in the study of conformal deformation of hyperbolic space: their existence implies the existence of complete Riemannian metrics conformally related to the classical metric of the hyperbolic space and with scalar curvature equal to \(-1\).
conformal deformation of hyperbolic space
scientific article; zbMATH DE number 1146376