Gelfand problem and Hemisphere rigidity

Abstract

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