We establish new results concerning existence and asymptotic behavior of entire, positive, and bounded solutions which converge to zero at infinite for the quasilinear equation − Δ p u = a ( x ) f ( u ) + λ b ( x ) g ( u ) ,   x ∈ ℝ N ,   1 < p < N , where f , g : [ 0 , ∞ ) → [ 0 , ∞ ) are suitable functions and a ( x ) , b ( x ) ≥ 0 are not identically zero continuous functions. We show that there exists at least one solution for the above-mentioned problem for each 0 ≤ λ < λ ⋆ , for some λ ⋆ > 0 . Penalty arguments, variational principles, lower-upper solutions, and an approximation procedure will be explored.