(7 in a closed subspace V of L2(R; R’) which is invariant under VF, where FE C1(2v, $3) is a strictly convex function and L is an unbounded self-adjoint operator on V with no essential spectrum. We assume that VF(0) = 0, so that u = 0 is a solution of (*). We also assume, without loss of generality, that F(0) = 0. Loosely speaking, we shall see that the number of nontrivial solutions of (*) is related to the number of eigenvalues of L “crossed by 2F(u)/luj’” as Iu] varies from 0 to x, provided that (*) is equivariant with respect to some group action, so that Lusternik-Schnirelman theory can be used. We apply this theory to the “dual action” associated with (*), which was introduced by Clarke and Ekeland [5] for Hamiltonian systems. The abstract framework and main results are presented in Section 2. In Section 3 we consider two applications. First we consider the nonlinear Dirichlet problem -Au = g(rc) in R u=O on aR, for which it is classical to use the Z,-action when g is odd [3]. When R is a disc in R2, the symmetry of the domain was used in [7] instead of a (possible) symmetry of the nonlinearity. In this case, a natural S’-action is provided by the rotations in R. In theorem 3 we extend the multiplicity result of [7]. Moreover the use of the dual action simplifies the proof. It is