We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: ( | f ′ | p − 2 f ′ ) ′ + β r f ′ + α f + ( f q ) ′ = 0 satisfying a specific decay rate: lim ⁡ r → ∞ r α / β f ( r ) = 0 with α : = ( p − 1 ) / ( p q − 2 p + 2 ) and β : = ( q − p + 1 ) / ( p q − 2 p + 2 ) . Here p > 2 and q > p − 1 . Such a solution arises naturally when we study a very singular self-similar solution for a degenerate parabolic equation with nonlinear convection term u t = ( | u x | p − 2 u x ) x + ( u q ) x defined on the half line [ 0 , + ∞ ) .