Singular solutions of semilinear elliptic equations with supercritical growth on Riemannian manifolds (Q6127264)

Let \((M^n,g)\) be an \(n\)-dimensional Riemannian model manifold containing a pole such the expression of the Riemannian metric \(g\) in spherical coordinates around the pole is given by \(ds^2=dr^2+\psi^2(r)d\theta^2,\) with \(r\in (0,R)\) and \(\theta\in \mathbb S^{n-1}\), the \((n-1)\)-dimensional unit sphere, where \(\psi\in C^2([0,R))\), the set of twice continuously differentiable functions on \([0,R)\), such that \(\psi(0)=\psi''(0)=0\) and \(\psi'(0)=1\), let \(f\in C^2([0,\infty))\), such that \(f(u)=-\Delta_gu\) in \((M^n,g)\setminus\{0\}\) and \(\displaystyle F(u)=\int_u^\infty\frac{ds}{f(s)}<\infty\) for \(u\ge u_0\) with some positive \(u_0\), and we assume that the following limit \(q=\displaystyle\lim_{u\to\infty}\left(\frac{[f'(u)]^2}{f(u)f''(u)}\right)\) is finite. Then, for \(r\in (0,r_0]\) for some \(r_0\in (0,R]\), the authors state that there is a unique singular solution \(u^*(r)\) of for the following ordinary differential equation: \[u''(r)+(n-1)\frac{\psi'(r)}{\psi(r)}u'(r)+f(u)=0\ for\ r\in(0,R).\] Furthermore, when \(r\) tends to zero, \(u^*(r)\) is provided explicitly in terms of \(F,\psi\), and \(q\), see Theorem 1.1 for the full statement.