Positive evanescent solutions of singular elliptic problems in exterior domains

We investigate the existence of positive solutions for the following class of nonlinear elliptic problems div(a(‖x‖)∇u(x)) + f (x, u(x))− (u(x))−α‖∇u(x)‖β + g(‖x‖)x · ∇u(x) = 0, where x ∈ Rn and ‖x‖ > R, with the condition lim‖x‖→∞ u(x) = 0. We present the approach based on the subsolution and supersolution method for bounded subdomains and a certain convergence procedure. Our results cover both sublinear and superlinear cases of f . The speed of decaying of solutions will be also characterized more precisely.