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By Abrashkin V.A.

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Here S(Sr) denotes the collection of all symmetric subsets of Sr. The critical values ck satisfy 1) C l 3ic 2 g---=ic f c gc k + 1 . 2) Ci = Mins /, CN = Maxs /. 3) If cj-cj+l = - - - = ci+p-1 then the set K,. = {x \ f(x) = c, /'(x) = Ax} has genus -y(Kc)^:p. The classical Ljusternik-Schnirelmann proof of this result was based on a different topological notion, that of category. 2; one simply constructs equivalent deformations. This classical result can be extended to functionals on Banach spaces which satisfy the Palais-Smale condition.

F5 and F6 are obtained as F5 = Fzz + Ffz, F6 = Fzz'l"1 + F 2 z n ~ 1 . 3. Let F = (Au + a\)z + (Cu+Dv + p\)zn~1 and assume Then codim F = 2 and (This is a slightly specialized version of a result due to Golubitsky and Schaeffer [37, Thm. ) Proof. We begin with the slightly more general case G = (Au + Bi) + aA)z + (Cu + Du + (3A)z n ~ 1 and then, for simplicity, restrict ourselves to the case B = 0. The generators of TF in this case are Now we claim that

12 (Bahri and Berestycki [3], [5]). 17) possesses infinitely many T-periodic solutions. Rabinowitz [54] had previously shown that systems of the type z = /VHz(t, z) possess at least one T-periodic solution if H(t, z) is a T-periodic bounded perturbation of a super quadratic Hamiltonian H(z). Bahri and Berestycki have derived similar results for the inhomogeneous boundary value problem In order to obtain their results, they applied a topological perturbation method developed by Bahri [2]. Consider the perturbed functional on the sphere ||u|| = l (||u|l is the Dirichlet norm of u).

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