Let Ω ⊂ R N , N ⩾ 2 , be a bounded domain. We consider the following quasilinear problem depending on a real parameter λ > 0 : ( P λ ) { − Δ N u = λ f ( u ) u > 0 } in Ω , u = 0 on ∂ Ω , where f ( t ) is a nonlinearity that grows like e t N / N − 1 as t → ∞ and behaves like t α , for some α ∈ ( 0 , N − 1 ) , as t → 0 + . More precisely, we require f to satisfy assumptions (A1)–(A5) in Section 1. With these assumptions we show the existence of Λ > 0 such that ( P λ ) admits at least two solutions for all λ ∈ ( 0 , Λ ) , one solution for λ = Λ and no solution for all λ > Λ . We also study the problem ( P λ ) posed on the ball B 1 ( 0 ) ⊂ R N and show that the assumptions (A1)–(A5) are sharp for obtaining global multiplicity. We use a combination of monotonicity and variational methods to show multiplicity on general domains and asymptotic analysis of ODEs for the case of the ball.