  We prove an existence result for a $p$-Laplacian problem set in the whole
Euclidean space and exhibiting a critical term perturbed by a singular,
convective reaction. The approach used combines variational methods, truncation
techniques, and concentration compactness arguments, together with set-valued
analysis and fixed point theory. De Giorgi's technique, a priori gradient
estimates, and nonlinear regularity theory are employed to get local
$C^{1,\alpha}$ regularity of solutions, as well as their pointwise decay at
infinity. The result is new even in the non-singular case, also for the
Laplacian.
