A sup + inf Inequality for Some Nonlinear Elliptic Equations Involving Exponential Nonlinearities

Let u be a solution of the equation −Δu = V(x) eu in a domain Ω ⊂ R2 where 0 ≤ a < V ≤ b and V is Lipschitz continuous. We prove that sup u can be controlled in terms of inf u. More precisely, supKu + infΩu ≤ C(a, b, K, Ω, ||∇ V||L∞) for any compact subset K⊂Ω. This extends an earlier result of Shafrir who obtained a similar conclusion when V≡1.