A three solution theorem for singular nonlinear elliptic boundary value problems

Abstract

We establish a three solution theorem for singular elliptic boundary value problems of the form −Δu=f(u)uβ in Ω, u=0 on ∂Ω where Ω is a bounded domain in RN, N≥1 with a smooth boundary ∂Ω. Here f:[0,∞)→[0,∞) is a C1 function in [0,∞) with f(0)>0 and β∈(0,1). In particular if there exist two pairs of sub-supersolutions (ψ1,ϕ1) and (ψ2,ϕ2) where ψ1≤ψ2≤ϕ1, ψ1≤ϕ2≤ϕ1 with ψ2≰ϕ2, and ψ2, ϕ2 are strict sub and supersolutions, then we establish existence of three solutions u1, u2 and u3 for the above boundary value problem such that u1∈[ψ1,ϕ2], u2∈[ψ2,ϕ1] and u3∈[ψ1,ϕ1]∖([ψ1,ϕ2]∪[ψ2,ϕ1]). Our results extend the work in [1] (and also in [17]) where such multiplicity results were discussed in the non-singular case β=0. Further our results strengthen the multiplicity results in [12] and [13] for such singular problems (β≠0), namely, establish an additional third solution. We also establish a sub-super solution theorem for infinite semipositone problems (i.e. the case when f(0)<0).