Generalized eigenvalue problems for -Laplacian with indefinite weight

This paper presents existence and non-existence results on a positive solution for quasilinear elliptic equations of the form − Δ r u − μ Δ r ⁎ u = λ m r ( x ) | u | r − 2 u in Ω with 1 < r ≠ r ⁎ < ∞ and μ > 0 , under Dirichlet boundary condition, where Ω is a bounded domain in R N and m r is a weight function in L ∞ ( Ω ) admitting sign-change. We show that existence and non-existence of a positive solution depend only on the relation between λ and the first eigenvalue of r-Laplacian with weight function m r , whence it is independent of the operator Δ r ⁎ and the parameter μ > 0 .