Asymptotically linear Dirichlet problem for the p-Laplacian

The existence of positive solutions for the following Dirichlet problem for the p-Laplacian (p>1): -pu(x)≡-div(∥▽u∥p-2▽u) = f(x,u), x∈Ω, u∈W01,p(Ω), where Ω is a bounded domain in RN (N≥1) with smooth boundary ∂Ω, and f is asymptotically linear with respect to u at infinity was studied. The existence of positive solutions are considered and the conditions imposed on f(x,t) are as follows: f(x,t)∈C(Ω×R); f(x,t)≥0, qqt≥0, x∈Ω and f(x,t)≡f(x,0)≡0, qqt≤0, x∈Ω; f(x,t)/tp-1 is nondecreasing in t>0, for x∈Ω; limt→0 f(x,t)/tp-1 = 0; limt→∞ f(x,t)/tp-1 = l (for some l>0) uniformly in x∈Ω. It follows from (f1)-(f3) that f(x,t)≤l gp-1 for ak x∈Ω, t≥0 and, for any ε>0, q∈(p,N p/(N-p)) if N>p or q∈(p,+∞) if 1≤N≤p, there exists Cε>0 such that f(x,t)≤εtp-1+Cεtq-1.