Existence and nonexistence of ground state solutions for elliptic equations with a convection term

We deal with the existence of positive solutions uu decaying to zero at infinity, for a class of equations of Lane–Emden–Fowler type involving a gradient term. One of the main points is that the differential equation contains a semilinear term σ(u)σ(u) where σ:(0,∞)→(0,∞)σ:(0,∞)→(0,∞) is a smooth function which can be both unbounded at infinity and singular at zero. Our technique explores symmetry arguments as well as lower and upper solutions.