Multiple boundary blow-up solutions for nonlinear elliptic equations

Abstract

We consider the problem Δu = λf(u) in Ω, u(x) tends to +∞ as x approaches ∂Ω. Here, Ω is a bounded smooth domain in R^N, N ≥ 1 and λ is a positive parameter. In this paper, we are interested in analysing the role of the sign changes of the function f in the number of solutions of this problem. As a consequence of our main result, we find that if Ω is star-shaped and f behaves like f(u) = u(u−a)(u−1) with ½ < a < 1, then there is a solution bigger than 1 for all λ and there exists λ₀ > 0 such that, for λ < λ₀, there is no positive solution that crosses 1 and, for λ > λ₀, at least two solutions that cross 1. The proof is based on a priori estimates, the construction of barriers and topological-degree arguments.