GRADIENT ESTIMATES FOR A NONLINEAR ELLIPTIC EQUATIONON COMPLETE NONCOMPACT RIEMANNIAN MANIFOLD

Abolarinwa, Abimbola (2018) GRADIENT ESTIMATES FOR A NONLINEAR ELLIPTIC EQUATIONON COMPLETE NONCOMPACT RIEMANNIAN MANIFOLD. Journal of Mathematical Inequalities, 12 (2). pp. 391-402.

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Abstract

Let(M,g)be ann-dimensional complete noncompact Riemannian manifold (withpossibly empty boundary). We derive local and global gradient estimates on positive solutionsu(x)to the following nonlinear elliptic equationΔu(x)+aus(x)+λ(x)u(x)=0,x∈M,whereaandsare constants,a∈R\{0},s>1andλ(x)is bounded onM. Our gradientestimates yield differential Harnack inequalities as an application. This paper extends results ofY. Yang [17]andJ.Li[11, Theorem 3.1].

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