Liouville theorems and gradient estimates for a nonlinear elliptic equation

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