Fourth-order nonlinear elliptic equations with lower order term and natural growth conditions

We prove the existence of weak solutions of the homogeneous Dirichlet problem related to a class of nonlinear elliptic equations whose prototype is ∑∣α∣=2Dα[∣D2u∣p−2Dαu]−∑∣α∣=1Dα[∣D1u∣q−2Dαu]+u[∣D1u∣q+∣D2u∣p]=f where ΩΩ is an open bounded subset of RNRN (N≥3N≥3) with sufficiently smooth boundary, u:Ω→Ru:Ω→R is the unknown function, Dhu={Dαu:∣α∣=h}, ∣Dhu∣=[∑∣α∣=h∣Dαu∣2]12, for h=1,2h=1,2, numbers pp, q∈[2,N[q∈[2,N[ and f∈L1(Ω)f∈L1(Ω).