Boundary Value Problems
Volume 2009 (2009), Article ID 845946, 16 pages
doi:10.1155/2009/845946

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 −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.