A variational principle associated with elliptic boundary value problems

We introduce a variational approach which can be applied to a large class of nonlinear elliptical equations. Assume that φ:R→R and ψ:R2→R are differentiable convex functions. We concern ourselves with problems of the form{Δ2u(x)+u(x)=φ′(u(x)),x∈Ω,β2u(x)=∇ψ(β1u(x)),x∈∂Ω, where Ω⊂Rn is an open, bounded domain with smooth boundary. The maps β1, β2 are boundary operators given by β1u=(u,∂u∂n) and β2u=(−∂Δu∂n,Δu) where n is the outward normal to ∂Ω. We shall show that solutions to this boundary value problem can coincide with critical points of a functional F defined byF(u)=∫Ωφ⁎(Δ2u+u)dx−∫Ωφ(u)dx+∫∂Ωψ⁎(β2u)dσ−∫∂Ωψ(β1u)dσ, where φ⁎ and ψ⁎ are Fenchel-Legendre dual of φ and ψ respectively. We then use these functionals to prove existence of nontrivial solutions to certain boundary value problems. This method offers advantages when compared to using the more standard Euler-Lagrange functional, in that solutions have greater regularity and nonlinear boundary conditions can be more easily dealt with.