On the existence of positive solutions of quasilinear elliptic boundary value problems

We establish the existence of a positive solution of a class of anisotropic singular quasilinear elliptic boundary value problems with certain nonlinearities. One example is: uauxx+ubuyy +lu+1a+r =0,u&vbm0; 6W=0. 1 Here Ω is a bounded convex smooth domain in R2, a ≥ b ≥ 0, λ \u3e 0, and r \u3e 0. ^ If 0 \u3c r \u3c 1 (sublinear case), then (1) has a solution for all λ \u3e 0. On the other hand, if r \u3e 1 (superlinear case), then there exists a positive constant λ* such that for each λ ∈ (0, λ*], (1) has a positive solution. ^ Finally, we present some numerical results for some open theoretical questions for these types of problems.