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

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