Singular anisotropic elliptic equations with gradient-dependent lower order terms

We prove the existence of weak solutions for a general class of Dirichlet anisotropic elliptic problems of the form Au + Phi (x, u, del u) = Psi (u, del u) + Bu + f on a bounded open subset Omega subset of R-N (N >= 2), where f is an element of L-1 (Omega) is arbitrary. Our models are Au = - Sigma(N)(j=1) partial derivative(j) (vertical bar partial derivative(j)u vertical bar(pj-2)partial derivative(j)u) and Phi (u, del u) = (1 + Sigma(N)(j-1) a(j)vertical bar partial derivative(j)u vertical bar(pj)) vertical bar u vertical bar m(-2) u, with m, p(j) > 1, a(j) >= 0 for 1 <= j <= N and Sigma(N)(k=1) (1/p(k)) > 1. The main novelty is the inclusion of possibly singlular gradient-dependent term Psi(u, del u) = Sigma(N)(j=1) vertical bar u vertical bar theta(-2)(j)u vertical bar partial derivative(j)u vertical bar(qj), where theta(j) > 0 and 0 <= q(j) < p(j) for 1 <= j <= N. Under suitable conditions, we prove the existence of solutions by distinguishing two cases: 1) for every 1 <= j <= N, we have theta(j) > 1 and 20 there exists 1 <= j <= N such that theta(j) <= 1. In the latter situation, assuming that f >= 0 a.c. in Omega, we obtain non-negative solutions for our problem.