A nonlinear eigenvalue problem for the periodic scalar \(p\)-Laplacian (Q2438941)

The authors consider the following nonlinear periodic eigenvalue problem \[ \begin{cases} -(|u'(t)|^{p-2}u'(t))'=\lambda|u(t)|^{p-2}u(t)-f(t,u(t)),\;t\in[0,b], \\ u(0)=u(b), u'(0)=u'(b),\end{cases} \] where \(1<p<\infty\), \(\lambda>0\), \(f(t,x)\) is a Carathéodory perturbation. They show that if \(\lambda>\lambda_1\), where \(\lambda_1\) is the first nonzero eigenvalue of the periodic scalar \(p\)-Laplacian, then the problem has at least three nontrivial solutions providing precise sign information for all the solutions.