Steady state solutions of a logistic equation with nonlinear boundary conditions

We consider the diffusive logistic equation with nonlinear boundary conditions δu + λu(1- u) = 0 x ∈Ω, α(u)(∂u/ ∂v) + (1-α(u))u = 0 x ∈ Ω, u 0 for i∈Ω, where Ω is a bounded domain in R^N with v its outer unit normal and A is a positive parameter. The boundary conditions of the equation considered are nonlinear, with the function α satisfying α([0,1]) C [0,1] and increasing. In this work we will study the case α(0) = 0, α′(0) > 0, which implies that, as A varies, the above equation has two continua of solutions, one having Dirichlet boundary conditions, and another one in which each solution is positive at the boundary. We show that the second continuum of solutions may contain infinitely many solutions for a fixed value of λ.