Positive Solutions For A Class Of Quasi-Linear Degenerate Elliptic Equations

In this paper we are interested in existence and nonexistence results for positive solutions of the following qusilinear elliptic problemwith the boundary conditionu = 0 on (?)Ω, (2)where Ω is a bounded open subset of R^N, N ≥ 1, p > 1, g : [0, ∞) → [0,∞) is a nonnegative continuous function, A is a real number.The solution of (1) is defined byDefinition A nonnegative function u : Ω → [0,∞) is said to be a solution of (1), if u ∈ W₀¹,^p(Ω)∩L¹(Ω) satisfying the following condition [H1]-[H2]:[H1] We haveg(u)|â–½u|^p ∈ L¹(Ω);[H2] For and φ ∈W₀¹,^p(Ω)∩L^∞(Ω), we haveSuch solutions have been discussed by many people, such as Ambrosetti, A.; Rabinowitz, P.H. [1], Celine Azizieh [2], Bensoussan, A.; Boccardo, L.; Murat, F. [3], Boccardo, L.; Gallouet, T.; Murat, F.A [4], Boccardo, L.; Gallouet, T.; Orsina, L. [5], Boccardo, L.; Murat, F.; Puel, J.P. [6], Brezis, H.; Nirenberg. L. [7], Murat, F.; Porretta, A. [8] and Pohozaev, S. [9].