Existence and Nonexistence of Positive Solutions for Perturbations of the Anisotropic Eigenvalue Problem

We consider a Dirichlet problem, which is perturbation of the eigenvalue problem for anisotropic p-Laplacian. assume that (p(z)−1)-sublinear, and we prove an existence nonexistence theorem positive solutions as parameter λ moves on semiaxis. also show smallest solution determine monotonicity continuity properties minimal map.