In this paper we study the structure of nonnegative nontrivial solutions of the following problem:-εΔ p u=f(u)inΩu>=0inΩ,u=0on Ωas ε->0 + , where Δ p u=div(|Du| p - 2 Du) with p>1. ε>0 is a small parameter and Ω is a bounded smooth domain in R N (N>=1). f is a class of logistic-type nonlinearities satisfying f(0)=f(z 1 )=f(z 2 )=0 with 0<z 1 <z 2 , f<0 in (0,z 1 ), f>0 in (z 1 ,z 2 ) and lim u - > 0 + f(u)/u p - 1 =-~. By virtue of the sub- and supersolution method, we prove that there are many nonnegative nontrivial solutions and they are spike-layered solutions. Moreover, the measure of each spike layer is estimated as ε->0 + .