Copyright © 2009 C. A. Santos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 −Δpu=a(x)f(u)+λb(x)g(u),  x∈ℝN,  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.