Existence and non-existence of positive solution for (p, q)-Laplacian with singular weights

Abstract

We use the Hardy-Sobolev inequality to study existence and non-existence results for a positive solution of the quasilinear elliptic problem -\Delta{p}u − \mu \Delta{q}u = \limda[mp(x)|u|p−2u + \mu mq(x)|u|q−2u] in \Omega driven by nonhomogeneous operator (p, q)-Laplacian with singular weights under the Dirichlet boundary condition. We also prove that in the case where μ > 0 and with 1 0 there exists an interval of eigenvalues for our eigenvalue problem.

Keywords

• Nonlinear eigenvalue problem
• (p, q)-Laplacian
• singular weight
• indefinite weight