Existence and multiplicity of nonnegative solutions for semilinear elliptic problems involving nonlinearities indefinite in sign

Let r,s∈]1,2[r,s∈]1,2[ and λ,μ∈]0,+∞[λ,μ∈]0,+∞[. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 00 boundary data for the semilinear elliptic equation −Δu=λus−1−ur−1−Δu=λus−1−ur−1 in Ω⊂RNΩ⊂RN, where N≥2N≥2. We prove that there exists a positive constant ΛΛ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λλ>Λ, λ=Λλ=Λ or λ<Λλ<Λ. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given.