A boundary blow-up problem with a nonlocal reaction

In this paper we deal with several kinds of boundary blow-up problems whose main feature is exhibiting a nonlocal reaction term which depends on the integral of some function of the concentration uu. As a representative of such problems we consider {Δu=λf(u)1+1∣Ω∣∫Ωg(u)in Ωu=∞on ∂Ω, where ΩΩ is a bounded smooth domain of RNRN and the boundary condition is understood as u(x)→∞u(x)→∞ when dist(x,∂Ω)→0dist(x,∂Ω)→0. We find necessary and sufficient conditions for the existence of positive solutions. In some cases, we also obtain uniqueness or multiplicity of solutions. Other classes of nonlocal dependence of the reaction term are also discussed.