A remark on a class of nonlinear eigenvalue problems

Let Ω⊂Rn be a bounded smooth domain and let λ1λ1 be the first eigenvalue of the problem {−Δu=λuin Ωu|∂Ω=0. In this paper, the following result is proved:Let f:R→R be a continuous function such that supξ∈R∫0ξf(t)dt=0. Put α=min{0,inf{ξ<0:f(ξ)<0}},β=max{0,sup{ξ>0:f(ξ)>0}},α=min{0,inf{ξ<0:f(ξ)<0}},β=max{0,sup{ξ>0:f(ξ)>0}}, and suppose that the restriction of ff to [α,β]∩R is Lipschitzian with Lipschitz constant LL.Then, for each λ∈[0,3λ1L[,0 is the only classical solution of the problem {−Δu=λf(u)in Ωu|∂Ω=0.