Solutions of quasilinear elliptic equations in R N decaying at infinity to a non-negative number

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We deal with existence of entire solutions for the quasilinear elliptic problem where 1 < p < N, N ≥ 3, ℓ ≥ 0, Δ p is the p-Laplacian operator, λ > 0 is a parameter, a : R N → (0, ∞) and f : (0, ∞) → (0, ∞) are suitable functions. When ℓ = 0, f is allowed to behave at 0 like . The potential a(x) will be required to decay to zero at infinity fast enough. Our technique explores variational principles, symmetry arguments as well as lower and upper solutions.