  We start by taking the analytical approach to discuss how the minimizer of
Yamabe functional provides constant scalar curvature and its relationship with
the Sobolev Space $W^{1,2}.$ Then, after demonstrating the importance of the
sphere $S^n$, with stereographic projection and dilation, we show that the
minimizer of Yamabe functional on standard sphere is obtained from a standard
round metric $\bar{g}$ by a conformal diffeomorphism, thus giving us the
constraint $\lambda (M) < \lambda(S^n),$ which leads us to the final theorem
that the Yamabe problem is solvable when $\lambda(M) < \lambda(S^n).$ For the
proof of this theorem, we adopt the approach of Concentration-Compactness.
