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In this paper we perform an extensive study of the existence, uniqueness (or multiplicity) and stability of nonnegative solutions to the semilinear elliptic equation  − Δu  =  λ u − u  p  in Ω, with the nonlinear boundary condition ∂u/∂ν  =  u  r  on ∂Ω. Here Ω is a smooth bounded domain of $${\mathbb{R}}^d$$ with outward unit normal ν, λ is a real parameter and p, r  >  0. We also give the precise behavior of solutions for large |λ| in the cases where they exist. The proofs are mainly based on bifurcation techniques, sub-supersolutions and variational methods.

Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary