Multiplicity and asymptotic behavior of nonnegative solutions for elliptic problems involving nonlinearities indefinite in sign

Let p,r,s,λ,μ be positive numbers with 1<r<s<p. The multiplicity of nonnegative solutions for the problem - Δpu= λus- 1-μur- 1 in Ω, u| ∂Ω=0, is investigated. We prove that there exist two positive constants Λ2 ,rs, Λrs with Λ2 ,rs< Λrs such that the above problem has at least two solutions, at least one solution or no solutions according to whether μλ- p-rp-s≤Λ2 ,rs, μλ- p-rp-s∈]Λ2 ,rs, Λrs] or μλ- p-rp-s> Λrs. We also study the asymptotic behavior of the solutions as μ→0 +.