Existence of solutions for semi-linear equations involving the p-Laplacian: the non coercive case

where Ω is a bounded smooth domain of R , 1 < p < N , p < q ≤ pN N−p := p , f and g belong to L∞, and λ ∈ R. By solution of (1.1), we mean a function u ∈ W 1,p 0 (Ω) satisfying (1.1) in the weak usual sense. In particular we shall study (1.1) considering the position ofλwith respect to the principal eigenvalue. Precisely, it is well known that the concept of “eigenvalue” and “eigenfunction” has been generalized by many authors to the quasi-linear setting of the p-Laplacian ∆p := div(|∇.|p−2∇.), in particular let us recall the works of Allegretto and Huang in [2], Anane in [3] and Lindqvist in [19]. We shall now state their definitions and the principal properties obtained in the works cited above.