Uniqueness of solutions of an elliptic singular boundary value problem (Q1269417)

The uniqueness of solutions of the following elliptic singular boundary value problem is proved: Let \((\widetilde M,\widetilde g)\) be a Riemannian manifold and \(M\) be a relatively compact domain in \(\widetilde M\) with \((n-1)\)-dimensional \(C^2\)-boundary, \(g:=\widetilde g|_M\), \(h\) be a nonnegative locally Hölder continuous function on \(M\) and \(r_{\partial M}\) be the distance function to the boundary \(\partial M\). If \(q>0\) and \(h\) satisfies the inequalities \(C_1(r_{\partial M})^\ell\leq h\leq C_2(r_{\partial M})^\ell\), \(\ell> -2\), \(C_1,C_2> 0\), then the problem \[ \Delta_g u= hu^q,\quad u>0\quad\text{on }M,\quad u\to+\infty\quad\text{as }r_{\partial M}\to 0 \] possesses a unique solution. Here \(\Delta_g\) is the Laplacian of \(g\), i.e. \(\Delta_g:= g^{ij}\nabla_{ij}\).