American Institute of Mathematical Sciences

We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. We also establish bounds on the periodic orbits that arise when the steady-state is unstable. This technique can be readily extended to apply to other scalar delay differential equations with negative feedback.

We prove that second order Hamiltonian systems \begin{document}$ -\ddot{u} = V_{u}(t,u) $\end{document} with a potential \begin{document}$ V\colon \mathbb{R} \times \mathbb{R} ^N\to \mathbb{R} $\end{document} of class \begin{document}$ C^1 $\end{document}, periodic in time and superquadratic at infinity with respect to the space variable have subharmonic solutions. Our intention is to generalise a result on subharmonics for Hamiltonian systems with a potential satisfying the global Ambrosetti-Rabinowitz condition from [14]. Indeed, we weaken the latter condition in a neighbourhood of \begin{document}$ 0\in \mathbb{R} ^N $\end{document}. We will also discuss when subharmonics pass to a nontrivial homoclinic orbit.

The \begin{document}$p$\end{document}-Laplacian operator \begin{document}$\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$\end{document} is not uniformly elliptic for any \begin{document}$p\in(1,2)\cup(2,\infty)$\end{document} and degenerates even more when \begin{document}$p\to \infty$\end{document} or \begin{document}$p\to 1$\end{document}. In those two cases the Dirichlet and eigenvalue problems associated with the \begin{document}$p$\end{document}-Laplacian lead to intriguing geometric questions, because their limits for \begin{document}$p\to\infty$\end{document} or \begin{document}$p\to 1$\end{document} can be characterized by the geometry of \begin{document}$\Omega$\end{document}. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general \begin{document}$p\in[1,\infty]$\end{document}. We report also on results concerning the normalized or game-theoretic \begin{document}$p$\end{document}-Laplacian

\begin{document}$\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$\end{document}

and its parabolic counterpart \begin{document}$u_t-\Delta_p^N u=0$\end{document}. These equations are homogeneous of degree 1 and \begin{document}$\Delta_p^N$\end{document} is uniformly elliptic for any \begin{document}$p\in (1,\infty)$\end{document}. In this respect it is more benign than the \begin{document}$p$\end{document}-Laplacian, but it is not of divergence type.