Existence and regularity to an elliptic equation with logarithmic nonlinearity

We study the nonlinear elliptic problem −Δu=χ{u>0}(logu+λf(x,u)) in Ω⊂Rn with u=0 on ∂Ω. The function is nondecreasing, sublinear and fu is continuous. For every λ>0, we obtain a maximal solution uλ?0 and prove its global regularity . There is a constant λ^∗ such that uλ vanishes on a set of positive measure for 0<λ<λ^∗, and uλ>0 for λ>λ^∗. If f is concave, for λ>λ^∗ we characterize uλ by its stability.