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

We study the solvability of the quasilinear elliptic problem of parameter s−Δpu=g(x,u)+sφ(x)in Ω,u=0on ∂Ω where Ω   is a smooth bounded domain in RNRN, φ⩾0φ⩾0, g(⋅,u)/|u|p−2ug(⋅,u)/|u|p−2u lies for u<0u<0 below the first eigenvalue of the p-laplacian and g   growths for u>0u>0 less than the lower Sobolev critical exponent p∗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∗s∗⩽s∗ such that the problem possesses no solutions if s>s∗s>s∗, it has at least one solution if s