We find the second positive radial solution for the following p -Laplacian problem: div ( | ∇ u | p − 2 ∇ u ) + K ( | x | ) u q = 0 in Ω , u | ∂ Ω = 0 , u ( x ) → μ > 0 as | x | → ∞ , where Ω = { x ∈ ℝ N : | x | > r 0 } , r 0 > 0 , N > p > 1 , K ∈ C ( Ω , ( 0 , ∞ ) ) and q > p − 1 . We also give some global existence results with respect to the parameter μ .