The Nehari manifold for a semilinear elliptic problem with the nonlinear boundary condition

Using the Nehari manifold and fibering maps, we prove the existence theorem of the nonlinear boundary problem − Δ u + u = | u | p − 2 u for x ∈ Ω ; ∂ u ∂ n = λ | u | q − 2 u for x ∈ ∂ Ω on a bounded domain Ω ⊆ R N . We discuss how the Nehari manifold changes as λ changes, and show how the existence of solutions depends on the properties of the Nehari manifold.