On an elliptic Kirchhoff-type problem depending on two parameters

摘要

In this paper, on a bounded domain Ω is contained in R~n, we consider a non-local problem of the typern{ -K (∫_Ω |▽u(x)|~2 dx) △u = λf(x, u) + μg(x, u) in Ω, u = 0 on partial deriv Ω.rnUnder rather general assumptions on K and f, we prove, in particular, that there exists λ~* > 0 such that, for each λ > λ~* and each Caratheodory function g with a sub-critical growth, the above problem has at least three weak solutions for every μ ≥ 0 small enough.