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 Sigma(vertical bar alpha vertical bar=2) D-alpha[vertical bar D(2)u|(p-2) D(alpha)u] - Sigma(vertical bar alpha vertical bar=1) D-alpha[vertical bar D(1)u|(q-2) D(alpha)u] + u[vertical bar D(1)u|(q) + vertical bar D(2)u|(p)] = f where Omega is an open bounded subset of R-N (N >= 3) with sufficiently smooth boundary, u : Omega -> R is the unknown function, D(h)u = {D(alpha)u :vertical bar alpha vertical bar = h}, vertical bar D(h)u vertical bar = [Sigma(vertical bar alpha vertical bar=h) vertical bar D(alpha)u vertical bar(2)](1/2), for h = 1, 2, numbers p, q is an element of [2, N[ and f is an element of L-1(Omega). (C) 2014 Elsevier Ltd. All rights reserved.