Non‐existence results for an eigenvalue problem involving Lipschitzian non‐linearities with non‐positive primitive

Let Ω ⊂ R N( N ⩾ 2 ) be a bounded smooth domain. In this paper, we prove that if either ∂ Ω has non‐negative mean curvature or Ω is an annulus, then, for each Lipschitzian function f : R → R such thatsup ξ ∈ R∫ 0 ξ f ( t ) d t = 0 , the problem− Δ u=f ( u )inΩ ,u =0on∂ Ωhas no non‐zero classical solutions.