Subharmonic solutions of nonlinear wave equations on \(S^n\)
The paper is concerned with the existence of \(2k\pi\)-periodic solutions of the wave equation (*)~\(u_{tt}-\Delta_{n}u = g(u,t)\), where \(\Delta_{n}\) is the Laplace-Beltrami operator on the sphere \(S^n\), \(g\) is \(2\pi\)-periodic in \(t\), superlinear in \(u\) as \(|u|\to\infty\) and \(g(u,0)\equiv 0\). Moreover, \(g_{u}(0,t)=\lambda\), where \(\lambda\) is not in the spectrum of \({\mathcal L} = \partial_{tt}-\Delta_{n}\) in \(L^2(S^n\times[0,2\pi])\). The authors show that if \(n\) is even, then for each large prime number \(k\) there exists a subharmonic solution \(u_{k}\) to (*) which has minimal period \(2k\pi\). Moreover, \(u_{k}\to 0\) in \(L^p(S^n\times[0,2k\pi])\) as \(k\to\infty\). If \(n\) is odd, the same results remain valid under more restrictive hypotheses on \(g\) (which are needed because in this case the operator \(\mathcal L\) has an eigenvalue of infinite multiplicity). The proof employs an abstract minimax principle of \textit{V. Benci, A. Capozzi} and \textit{D. Fortunato} [Ann. Mat. Pura Appl. (4) 143, 1-46 (1986; Zbl 0632.34036)]. That the solutions \(u_{k}\) have minimal period \(2k\pi\) and not \(2\pi\) follows from the fact that \(u_{k}\to 0\) in \(L^p\) and 0 is an isolated \(2\pi\)-periodic solution of (*). Some results on the existence of multiple subharmonics are also proved in this paper for \(g\) odd in \(u\) or independent of \(t\). The functional corresponding to the problem is then respectively \({\mathbb Z}_{2}\)- and \(S^1\)-invariant.
scientific article; zbMATH DE number 1383225
scientific article