GRADIENT ESTIMATES FOR A NONLINEAR ELLIPTIC EQUATION ON COMPLETE NONCOMPACT RIEMANNIAN MANIFOLD

Let (M,g) be an n-dimensional complete noncompact Riemannian manifold (with possibly empty boundary). We derive local and global gradient estimates on positive solutions u(x) to the following nonlinear elliptic equation Delta u(x) + au(3)(x) + lambda(x)u(x) = 0, x is an element of M where a and s are constants, a is an element of R \ {0}, s > 1 and lambda(x) is bounded on M. Our gradient estimates yield differential Harnack inequalities as an application. This paper extends results of Y. Yang [17] and J. Li [11, Theorem 3.1].