A remark on Ricceri's conjecture for a class of nonlinear eigenvalue problems

作者：Fan Xianling^*

来源：Journal of Mathematical Analysis and Applications, 2009, 349(2): 436-442.

DOI：10.1016/j.jmaa.2008.08.051

• Nonlinear eigenvalue problem
• Elliptic equation
• Lipschitz condition

Consider the eigenvalue problem(P(lambda(f) over tilde)): -Delta u =lambda f(x,u) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R(N). Denote by (C) over tilde (L) the set of all Caratheodory functions f : Omega x R -> R such that for a.e. x epsilon Omega, f(x, .) is Lipschitzian with Lipschitz constant L, F(x,0) = 0 abd sup(E epsilon R)integral(E)(0) f(x, t) dt = 0, and denote by Lambda((f) over tilde) (resp. Lambda(W)((f) over tilde)) the set of lambda > 0 such that (P(lambda(f) over tilde)) has at least one nonzero classical (resp, weak) solution. Let lambda(1) be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that inf(f epsilon(C) over tildeL) inf Lambda((f) over tilde) = inf(f epsilon(C) over tildeL) inf Lambda(W)((f) over tilde) =3 lambda 1/L and {inf Lambda((f) over tilde) verti

• 出版日期2009-1-15
• 单位兰州城市学院; 兰州大学