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 −Δu+u=|u|p−2uforx∈Ω;∂u∂n=λ|u|q−2uforx∈∂Ω on a bounded domain Ω⊆RNΩ⊆RN. We discuss how the Nehari manifold changes as λλ changes, and show how the existence of solutions depends on the properties of the Nehari manifold.