Liouville theorems and Harnack inequalities for Allen–Cahn type equation

We first give a logarithmic gradient estimate for the local positive solutions of Allen–Cahn equation on the complete Riemannian manifolds with Ricci curvature bounded below. As its natural corollary, Harnack inequality and a Liouville theorem for classical positive solutions are obtained. Later, we consider similar estimate under integral curvature condition and generalize previous results to a class nonlinear equations which contain some classical elliptic equations such as Lane–Emden equation, static Whitehead–Newell equation and static Fisher–KPP equation. Last, we briefly generalize them to equation with gradient item under Bakry–Émery curvature condition.