Multiplicity of solutions of variational systems involving φ-Laplacians with singular φ and periodic nonlinearities

Using a Lusternik-Schnirelman type multiplicity result for some indefinite functionals due to Szulkin, the existence of at least $n+1$ geometrically distinct T-periodic solutions is proved for the relativistic-type Lagrangian system (φ(q'))' + ∇qF(t,q) = h(t), where φ is an homeomorphism of the open ball Bₐ ⊂ Rn onto Rn such that φ(0) = 0 and φ = ∇ Φ, F is Tⱼ-periodic in each variable qⱼ and h ∈ Lˢ(0,T;Rn) $(s > 1)$ has mean value zero. Application is given to the coupled pendulum equations (q'ⱼ√1 - \|q\|²)' + Aⱼ sin qⱼ = hⱼ(t) (j = 1,…,n). Similar results are obtained for the radial solutions of the homogeneous Neumann problem on an annulus in Rn centered at $0$ associated to systems of the form ∇ · (∇ wᵢ√1 - ∑ⱼ₌₁ⁿ \|∇ wⱼ\|²) + ∂wⱼ G(\|x\|,w) = hᵢ(\|x\|), (i = 1,…,n), involving the extrinsic mean curvature operator in a Minkovski space.