Anisotropic elliptic equations with gradient-dependent lower order terms and L1 data

We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as Au + phi(x, u, del u) =B3u + f in Omega, where Omega is a bounded open subset of R-N and f is an element of L-1(Omega) is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator A, the prototype of which is Au = - Sigma(N)(j=1) partial derivative(j) (vertical bar partial derivative(j)u vertical bar(pj-2)partial derivative(j)u) with p(j) > 1 for all 1 <= j <= N and Sigma(N)(j=1)(1/p(j)) > 1. As a novelty in this paper, our lower order terms involve a new class of operators B such that A - B is bounded, coercive and pseudo-monotone from W-0(1 (p) over right arrow) (Omega) into its dual, as well as a gradient-dependent p nonlinearity Phi with an "anisotropic natural growth " in the gradient and a good sign condition.