Positive solutions for infinite semipositone problems with falling zeros

We consider the positive solutions to the singular problem equation(P){−Δu=au−f(u)−cuαin Ωu=0on ∂Ω where 0<α<10<α<1,a>0a>0 and c>0c>0 are constants, ΩΩ is a bounded domain with smooth boundary and f:[0,∞)→Rf:[0,∞)→R is a continuous function. We assume that there exist M>0M>0,A>0A>0,p>1p>1 such that au−M≤f(u)≤Aupau−M≤f(u)≤Aup, for all u∈[0,∞)u∈[0,∞). A simple example of ff satisfying these assumptions is f(u)=upf(u)=up for any p>1p>1. We use the method of sub–supersolutions to prove the existence of a positive solution of (P) when a>2λ11+α and cc is small. Here λ1λ1 is the first eigenvalue of operator −Δ−Δ with Dirichlet boundary conditions. We also extend our result to classes of infinite semipositone systems.