Multiple periodic solutions of autonomous semilinear wave equations (Q1337623)

Consider the semilinear wave equation \(u_{tt} - u_{xx} = g(u)\) subject to the boundary and the periodicity conditions \(u(0,t) = u(\pi,t) = 0\) and \(u(x,t + T) = u(x,t)\) \((T\) is a rational multiple of \(\pi)\). Denote by \(\{\lambda_ h\}\) the set of those eigenvalues of the wave operator which are of the form \(j^ 2 - 4 \pi^ 2 k^ 2/T^ 2\), \(j_ k \geq 0\), \(j\) odd, \(k\) even. The main result of the paper asserts that under suitable hypotheses this problem has at least as many nonconstant in time (and geometrically distinct) solutions as is the number of \(\lambda_ h\)'s which \(g(s)/s\) crosses as \(s\) goes from 0 to infinity. The proof uses a critical point theorem for \(S^ 1\)-invariant functionals established by the authors in [Multiple periodic solutions of asymptotically linear Hamiltonian systems. Q. Semin. Mat. Brescia 8/93 (1993)] and an argument due to \textit{J. M. Coron} [Math. Ann. 262, 273-285 (1983; Zbl 0503.35012)] which allows to remove the usual assumption that \(g\) is monotone.