On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent
نویسندگان
چکیده
We consider the nonlinear eigenvalue problem −div ( |∇u|∇u ) = λ|u|u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R with smooth boundary and p, q are continuous functions on Ω such that 1 < infΩ q < infΩ p < supΩ q, supΩ p < N , and q(x) < Np(x)/ (N − p(x)) for all x ∈ Ω. The main result of this paper establishes that any λ > 0 sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland’s variational principle. 2000 Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02.
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