Critical-point theory with applications to asymptotically linear wave and beam equations

Introduction.We consider a class of functionals Φ ∈ C 1 (E, R), defined on a Hilbert space E, of the formwhere the sum u = u + +u 0 +u -corresponds to an orthogonal decompositionIn our applications all subspaces E + , E 0 , E -are infinite-dimensional.The nonquadratic part Ψ is defined on a Hilbert space H containing E as a dense subspace.The inclusions E ± → H are compact, and E 0 is a closed subspace of H.The functional Ψ ∈ C 1 (H, R) is convex but not necessarily strictly convex.We develop some critical-point theory for this class of functionals which is essentially based on the fact that we can control the level Φ(u) of a weak limit u of certain Palais-Smale sequences.Simple examples show that Φ need not satisfy the Palais-Smale condition.In fact, there may exist bounded Palais-Smale sequences without a convergent subsequence.We also have Lusternik-Schnirelmann-type results for even Φ.The abstract theory is applied to prove the existence of a forced vibration for a nonlinear wave equation or a nonlinear beam equation.These equations have the formfor x ∈ Ω := (0, π), t ∈ R;Bu(x, t) = 0 for x ∈ ∂Ω, t ∈ R; u(x, t + 2π) = u(x, t) for x ∈ Ω, t ∈ R.