We consider an elliptic problem of Ambrosetti–Prodi type involving critical Sobolev exponent. We prove that this problem has solutions blowing up near the boundary of the domain.

In this paper, we study the existence and limiting behavior of the mountain pass solutions of the elliptic problem —Δu = λf(u — q(x)) in Ω C ℝ²; u 0 on ∂Ω, where q is a positive harmonic function. We show that the 'vortex core' Aλ = {x ∈ Ω: uλ (x) > q(x)} of the solution uλ, shrinks to a global minimum point of q on the boundary ∂Ω as λ → + ∞. Furthermore, we show that for each strict local minimum x₀ point of q(x) on the boundary ∂Ω, there exists a solution uλ whose vortex core shrinks to this strict local minimum point x₀ as λ → +∞.