Symmetry for elliptic equations in a half-space without strong maximum principle

Abstract

For a wide class of nonlinearities f (u) satisfying f (u) > 0 in (0, a) and f(u) < 0 in (a,∞), but not necessarily Lipschitz continuous, we study the quasi-linear equation -Δ pu = f(u) in T, u│∂T = 0, where T = {x = (x₁,x₂,...., xN )ϵ ℝ^2 RN :x₁ > 0}≽ with N > 2. By using a new approach based on the weak maximum principle, we show that any positive solution on T must be a function of x1 only. Under our assumptions, the strong maximum principle does not hold in general and the solution may develop a flat core; our symmetry result allows an easy and precise determination of the flat core.