Gradient and Hessian estimates for an elliptic equation on smooth metric measure spaces (Q2075212)

Given a metric measure space \((M^n,g,e^{-f}dV)\), the author considers positive solutions to \[\Delta_f u=pu^a\,;\tag{1} \] where \(a>0\), \(\Delta_f u\) is the usual \(f\)-Laplace operator: \[ \Delta_f u:=\Delta u-(\nabla u,\nabla f)\, , \] while \(f\) and \(p\) are smooth functions. In particular, in the case \(p\equiv 0\), Equation (1) becomes the \(f\)-harmonic equation. Under the assumptions that the Bakri-Emery-Ricci tensor is bounded from below the author proves some gradient (Theorem 1.1) and Hessian estimates (Theorem 1.4). As a corollary of these estimates, in the case the metric \(g\) is \((m,n)\)-quasi Einstein, he obtains a Liouville-type result for positive \(f\)-harmonic functions such that \(u(x)=o(\sqrt{r(x)})\) near infinity (Corollary 1.3). In addition, in Theorem 1.6 and in Corollary 1.7, the author proves the analogue of Shi's gradient and higher-derivative estimates for positive solutions to (1).