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 (phi(q'))' + del F-q(t,q) - h(t), where phi is an homeomorphism of the open ball B-a subset of R-n onto R-n such that phi(0) = 0 and phi = del Phi, F is T-j-periodic in each variable q(j) and h is an element of L-s(0, T; R-n) (s > 1) has mean value zero. Application is given to the coupled pendulum equations (q(j)'/root 1 - parallel to q parallel to(2))' + A(j) sin q(j) = h(j)(t) (j = 1, ... , n). Similar results are obtained for the radial solutions of the homogeneous Neumann problem on an annulus in R-n centered at 0 associated to systems of the form del . (del w(i)/root 1 - Sigma(n)(j=1)parallel to del w(j)parallel to(2)) + partial derivative wjG(parallel to x parallel to, w) = h(i)(parallel to x parallel to), (i = 1, .... , n), involving the extrinsic mean curvature operator in a Minkovski space.