p(x)-Laplacian with indefinite weight

By K. Kefi. 2011

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We consider the eigenvalue problem −div ( |∇u|p(x)−2∇u ) = λV (x)|u|q(x)−2u, in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R N , λ > 0, p, q are continuous functions on Ω and V is a given function in a generalized Lebesgue space Ls(x)(Ω) such that V > 0 in an open set Ω0 ⊂ Ω, where |Ω0| > 0. We prove under appropriate conditions on the functions p, q and s 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6ff334348753705637d55401b7e686f1b9d758bf