Multiple nonnegative solutions for an elliptic boundary value problem involving combined nonlinearities

Let p > 1 , q > p , s ∈ ] 1 , p [ , r ∈ ] 1 , s [ and let λ , μ be real positive parameters. We establish a multiplicity result for the problem − Δ p u = λ u s − 1 − μ u r − 1 + u q − 1 in Ω , u ∣ ∂ Ω = 0 by means of variational methods. In particular, for λ and μ small enough, the existence of at least two nonzero and nonnegative solutions is obtained. If q is less than the critical exponent, we also get the existence of a third nonzero and nonnegative solution.