Abstract:A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern-Simons Higgs model and have been studied recently by various authors. IntroductionLet (M, g) be a compact Riemann surface without boundary, V be a positive function on M , W be a function with M W dv g = 1. Throughout the paper dv g denotes the volume element of g, g denotes the Laplace Beltrami operator with respect to g. For λ ∈ R, we seek a solution ofClearly M W dv g = 1 is a necessary condition for (E u ) λ to have a solution. If we set ξ = u − log M V e u dv g for a solution of (E u ) λ , then ξ satisfies

In this paper we derive global W 1,∞ and piecewise C 1,α estimates for solutions to divergence form elliptic equations with piecewise Hölder continuous coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces.

In this paper we mainly introduce a min-max procedure to prove the existence of positive solutions for certain semilinear elliptic equations in \mathbb R^N .

Let Ω be a bounded open set in R n with C 2,α boundary, n ≥ 2, 0 < α < 1, D 1 and D 2 be two bounded strictly convex open subsets in Ω with C 2,α boundaries which are ε apart and far away from ∂Ω, i.e.where κ 0 , r 0 > 0 are universal constants independent of ε. We denoteGiven ϕ ∈ C 2 (∂Ω), consider the following scalar equation with Dirichlet boundary condition: