Some characterizations of ?-Einstein solitons

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