Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary

In this paper we consider the problem {-Delta u = u(q alpha)vertical bar del u vertical bar(q) +lambda f(x) in Omega u = 0 on partial derivative Omega , (P) where Omega subset of R(N) is a bounded domain, 1 < q <= 2, alpha is an element of R and f >= 0. We prove that: (1) If q alpha < -1, then problem (P) has a distributional solution for all f is an element of L(1)(Omega), and all lambda > 0. (2) If -1 <= q alpha < 0, then problem (P) has a solution for all f is an element of L(s)(Omega), where s > N/q if N >= 2, and without any restriction on lambda. (3) If q = 2 and -1 <= q alpha < 0 then problem (P) has infinitely many solutions under suitable hypotheses on f. (4) If 0 <= q alpha and f is an element of L(1)(Omega) satisfies lambda 1 (f) = inf (phi is an element of W01,2(Omega)) integral(Omega)vertical bar del phi vertical bar(2)dx/integral(Omega)f phi(2)dx > 0, then problem (P) has a positive solution if 0 < lambda < lambda(1)(f) and no positive solution for large lambda. (C) 2010 Elsevier Ltd. All rights reserved.