A remark on a class of nonlinear eigenvalue problems Original Research Article

Let View the MathML sourceΩ⊂Rn be a bounded smooth domain and let λ1λ1 be the first eigenvalue of the problem View the MathML source{−Δu=λuin Ωu|∂Ω=0. Turn MathJax on In this paper, the following result is proved: Let View the MathML sourcef:R→R be a continuous function such that View the MathML sourcesupξ∈R∫0ξf(t)dt=0. Turn MathJax on 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}}, Turn MathJax on and suppose that the restriction of ff to View the MathML source[α,β]∩R is Lipschitzian with Lipschitz constant LL. Then, for each View the MathML sourceλ∈[0,3λ1L[,0 is the only classical solution of the problem View the MathML source