Subharmonic solutions of the prescribed curvature equation (Q2800063)

The authors study the existence of subharmonic solutions of the prescribed curvature equation NEWLINENEWLINE\[NEWLINE -(u'/\sqrt{1+u{'}^{2}})'=f(t,u), NEWLINE\]NEWLINE NEWLINEwhere \(f:{\mathbb R}\times{\mathbb R}\to{\mathbb R}\) is a continuous (or Carathéodory) function, which is \(T\)-periodic in its first variable. NEWLINENEWLINEThey first prove that, when NEWLINENEWLINE\[NEWLINE \lim_{s\to0}\frac{f(t,s)}{s}=0\quad\text{uniformly in } t\in[0,T] NEWLINE\]NEWLINE NEWLINEtogether with a sign condition, then there exists a sequence \((u_k)_k\) of classical subharmonic solutions such that \(\|u_k\|_{C^1}\to 0\), and whose minimum periods diverge. The proof makes use of a generalized version of the Poincaré-Birkhoff Theorem.NEWLINENEWLINEThen, they consider the case when NEWLINENEWLINE\[NEWLINE \lim_{|s|\to\infty}f(t,s)=0\quad\text{uniformly in }t\in[0,T]NEWLINE\]NEWLINE NEWLINEtogether with a kind of Ahmad-Lazer-Paul condition, and they prove in this case the existence of a sequence \((u_k)_k\) of bounded variation subharmonic solutions having arbitrarily large amplitudes, whose minimum periods diverge. NEWLINEThe proof uses variational arguments based on the estimate of critical levels obtained through a saddle point theorem.