A one side superlinear Ambrosetti–Prodi problem for the Dirichlet p-laplacian

We study the solvability of the quasilinear elliptic problem of parameter s − Δ p u = g ( x , u ) + s φ ( x ) in Ω , u = 0 on ∂ Ω where Ω is a smooth bounded domain in R N , φ ⩾ 0 , g ( ⋅ , u ) / | u | p − 2 u lies for u < 0 below the first eigenvalue of the p-laplacian and g growths for u > 0 less than the lower Sobolev critical exponent p ∗ . We combine topological methods via upper and lower solutions and blow-up techniques to get a priori bounds to prove the following result of Ambrosetti–Prodi type: there exists s ∗ ⩽ s ∗ such that the problem possesses no solutions if s > s ∗ , it has at least one solution if s < s ∗ , and at least two solutions if s < s ∗ . We prove also that s ∗ = s ∗ in some cases.