Given a nonnegative bounded Radon measure mu on Omega C R-N, we discuss the existence or nonexistence of minima of infinite energy (so-called weak minima, T-minima, renormalized minima) for functionals like J(v) = integral(Omega) alpha(x, v)vertical bar Delta v vertical bar(p) dx - integral(Omega) v d mu where p > 1. In most of our results, alpha(x, s) is coercive. According to the behavior of s -> alpha(x, s) at infinity, existence or nonexistence of such minima is proved, and the convergence of approximating minima of regularized functionals is studied. Differences arise whether the measure charges or not sets of null p-capacity and/or alpha(x, s) blows-up at infinity. Lastly, some results are proved when alpha(x, s) degenerates at infinity.