Harnack inequalities for Yamabe type equations

Liouville type theorems and Harnack type inequalities for semilinear elliptic equations

Interpolating between Li-Yau and Chow-Hamilton Harnack inequalities along the Yamabe flow

In this paper, we establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality to the constrained trace Chow-Hamilton Harnack inequality for a nonlinear parabolic equation with respect to evolving metrics related to the Yamabe flow on the n-dimensional complete manifold. M.S.C. 2010: 53C44, 53C25.

Harnack inequalities and Bôcher-type theorems for conformally invariant fully nonlinear degenerate elliptic equations

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

On Harnack Inequalities and Singularities of Admissible Metrics in the Yamabe Problem

In this paper we study the local behaviour of admissible metrics in the kYamabe problem on compact Riemannian manifolds (M, g0) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g0) is not conformally equivalent to the unit sphere S and that the set of all such metrics is compact. When (M,g0) is the unit sphere we prove there is a unique ad...

Yamabe type equations on finite graphs

Let G = (V, E) be a locally finite graph, Ω ⊂ V be a bounded open domain, ∆ be the usual graph Laplacian, and λ1(Ω) be the first eigenvalue of −∆ with respect to Dirichlet boundary condition. Using the mountain pass theorem of Ambrosette and Rabinowitz, We prove that if α < λ1(Ω), then for any p > 2, there exists a positive solution to  −∆u − αu = |u| p−2u in Ω◦, u = 0 on ∂Ω, where Ω◦ and...