Existence Results for Singular Anisotropic Elliptic Boundary-value Problems

Abstract

We establish the existence of a positive solution for anisotropic singular quasilinear elliptic boundary-value problems. As an example of the problems studied we have uα uxx+ ub uyy + λ (u + 1)α+r = 0 with zero Dirichlet boundary condition, on a bounded convex domain in ℝ2. Here 0 ≤ b ≤ α, and λ, r are positive constants. When 0 < r < 1 (sublinear case), for each positive λ there exists a positive solution. On the other hand when r > 1 (superlinear case), there exists a positive constant λ* such that for λ in (0, λ*) there exists a positive solution, and for λ* < λ there is no positive solution.