Abstract
Let ω be a bounded domain in ℝn−1 with smooth boundary, u+,u−∈ℝ, a>0, and let u∈Wloc2,n((−a,a)×ω)∩C1([−a,a]×ω¯) satisfy −Δu+c(x1)ux1=f(x1,u) and ux1≥0 in (−a,a)×ω, u=u± on {±a}×ω and ∂u/∂v=0 on (−a,a)×∂ω, where c is bounded and nonincreasing and f is continuous and nondecreasing in x1. We prove that u is a function of x1 only. The same result is shown for a related problem in the infinite cylinder ℝ×ω. The proofs are based on a rearrangement inequality.