Multiplicity of solutions for nonlinear differential equations via non-convexity

Very recently, the following general result has been established: let E be a reflexive real Banach space, H a real Hilbert space, I a C^1 functional on E, J a C^1 operator from E to H. Assume that J(X) is not convex and that, for each y in H, the functional I(.)+ is weakly lower semicontinuous and coercive. Then, for every convex dense subset S of H, there exists v in S such that the functional I(.)+ has at least two critical points. The proposed thesis deals with some applications of the above theorem to: a) Dirichlet problem for second-order semilinear ellipltic equations; b) periodic solutions for Lagrangian systems of relativistic oscillators.