Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources

In this paper we deal with elliptic and parabolic quasilinear problems with a singular quadratic lower order term depending on the gradient, i.e., with the equations -div(M(x)del u) + B vertical bar del u vertical bar(2)/u(theta) = u(r) + f in Omega and u' -div(M(x, t)del u) + B vertical bar del u vertical bar(2)/u(theta) = u(r) in Omega x (0, T), u(x, 0) = u(0)(x) in Omega, with Omega a bounded open set of R-N, T > 0, M a bounded measurable uniformly elliptic matrix, B > 0, 0 < theta < 1 and 0 < r < 2 - theta. We will prove existence result for solutions under various assumptions on f and the initial datum u(0). Note that the elliptic equation is strongly related with the Euler-Lagrange equation of some integral functionals.