A remark on Ricceri's conjecture for a class of nonlinear eigenvalue problems |
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Authors: | Xianling Fan |
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Institution: | a Department of Mathematics, Lanzhou City University, Lanzhou 730070, Gansu, PR China b Department of Mathematics, Lanzhou University, Lanzhou 730000, PR China |
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Abstract: | Consider the eigenvalue problem : −Δu=λf(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN. Denote by the set of all Carathéodory functions f:Ω×R→R such that for a.e. x∈Ω, f(x,⋅) is Lipschitzian with Lipschitz constant L, f(x,0)=0 and , and denote by (resp. ) the set of λ>0 such that has at least one nonzero classical (resp. weak) solution. Let λ1 be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that and . Our result is a positive answer to Ricceri's conjecture if use f(x,u) instead of f(u) in the conjecture. |
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Keywords: | Nonlinear eigenvalue problem Elliptic equation Lipschitz condition |
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