  In this article we have showed that a gradient $\rho$-Einstein soliton with a
vector field of bounded norm and satisfying some other conditions is isometric
to the Euclidean sphere. Later, we have proved that a non-trivial complete
gradient $\rho$-Einstein soliton with finite weighted Dirichlet integral and
certain restriction on Ricci curvature must be of constant scalar curvature and
steady Ricci flat. Finally, we have proved that a non-shrinking or
non-expanding gradient traceless Ricci soliton possessing some conditions must
be steady.
