Liouville theorems and gradient estimates for a nonlinear elliptic equation

In this paper we establish gradient estimates for positive solutions to the equation (0.1)△fup=−λu on any smooth metric measure space whose m-Bakry–Emery curvature is bounded from below by −(m−1)K with K≥0. These estimates imply Liouville theorems for (0.1). When p→1, our main theorem reduces to the gradient estimate of Wang (2010) [9]. As applications, several Harnack inequalities are obtained.