Quasilinear Elliptic Problems with General Growth and Nonlinear Term Having Singular Behavior

In this paper we deal with the following quasilinear elliptic problems {-Delta(p)u + g(u) q(d(x)) = +/-vertical bar del u vertical bar(v) + lambda f(x, u), in Omega, u >= 0 in Omega, u = 0 on partial derivative Omega, where Omega subset of R(N) is a bounded regular domain, d(x) = dist(x, partial derivative Omega), p > 1, lambda > 0 and 0 < v <= p. Moreover, g, q and f are nonnegative functions verifying suitable hypotheses. The main goal of this work is to analyze the interaction between the functions d, g and the gradient term to get existence and nonexistence results. The singular behavior of g near 0 will provide a new phenomenon that does not appear in the case where the absorption term g(u) q(d(x)) is dropped or substituted by a regular one.