Generalized eigenvalue problems for (p,q)-Laplacian with indefinite weight

This paper presents existence and non-existence results on a positive solution for quasilinear elliptic equations of the form−Δru−μΔr⁎u=λmr(x)|u|r−2uin Ω with 10, under Dirichlet boundary condition, where Ω is a bounded domain in RN and mr 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 mr, whence it is independent of the operator Δr⁎ and the parameter μ>0.