Entire Solutions for a Quasilinear Problem in the Presence of Sublinear and Super-Linear Terms

Abstract

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 λb 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.