On the Existence of Positive Solutions of Quasilinear Elliptic Boundary Value Problems

We establish the existence of positive solutions to a class of quasilinear anisotropic problems which have either sublinear or superlinear nonlinearity. With a, b nonnegative constants and α, β positive constants, one example is If b−a<1 (sublinear case), then for each λ [0, ∞), (1) has a solution. On the other hand, if b−a>1 (superlinear case), then there exists a λ*>0 such that for 0 λ<λ*, (1) has at least one solution, and for λ>λ* no solution exists.