Gelfand problem and Hemisphere rigidity

We give an interpretation of the hemisphere rigidity theorem Hang-Wang in framework Gelfand problem. More precisely, showed that for a metric $ g conformal to standard g_0 on S^{n}_{+} with R\geq n(n-1) and whose boundary coincides g_0|_{\partial S^{n}_{+}} $, then = $. This is related classical problem, which investigates -\Delta u \lambda g(u) certain nonlinearity bounded region \Omega \subset \mathbb{R}^n subject Dirichlet condition. It well-known there exists extremal \lambda^{*} such \lambda>\lambda^{*} above equation does not admit any solution. Interestingly, Hang-Wang's yields precise value e^{2u} when n 2 (1+u)^{\frac{n+2}{n-2}} n\geq 3 attempt generalize under Q curvature lower bound fit this into fourth order problem bi-Laplacian suitable nonlinearity.