Multiplicity Of Positive Solutions For P(x)-Laplace Dirichlet Problems On A Bounded Domain In R~N

We show that the p (x)-Laplace Dirichlet problemwhere 1 < inf_Ωp(x)≤p (x)≤sup_Ωp (x) <∞, f(x, u) is a Carath(?)odory function,Ωis a bounded domain in R^N, with the boundary of class C², admits two positive solutions u₀, u₁ in W₀^1,(p(x))(Ω) such that 0 < u₀≤u₁ inΩ, while u₀ is a local minimizer of the associated Euler-Lagrange functional and u₁ is of Mountain Pass type.