In this paper, a class of damped vibration problems with impulsive effects is considered. An existence result is obtained by using the variational method and the critical point theorem due to Brezis and Nirenberg. The obtained result is also valid and new for the corresponding second-order impulsive Hamiltonian system. Finally, an example is presented to illustrate the feasibility and effectiveness of the result.

The existence of global solutions and the phenomenon of blow-up of a solution in finite time for a recently derived shallow water equation are studied. We prove that the only way a classical solution could blow-up is as a breaking wave for which we determine the exact blow-up rate and, in some cases, the blow-up set. Using the correspondence between the shallow water equation and the geodesic flow on the manifold of diffeomorphisms of the line endowed with a weak Riemannian structure, we give sufficient...

Summary: We give an introduction to the Skyrme model from a mathematical point of view. Hereby, we show that it is difficult to solve the field equation even by means of the classical ansatz, the so-called hedgehog ansatz. Our main result is an extended existence proof for solutions of the field equation in the hedgehog ansatz.

The purpose of this paper is to study the existence and multiplicity of a periodic solution for the non-autonomous second-order system $$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left(\right|\dot{u}\left(t\right){|}^{p-2}\dot{u}\left(t\right))=\nabla F(t,u\left(t\right)),\text{a.e.}t\in [0,T],\\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0,\\ \Delta {\dot{u}}^i\left(t_j\right)={\dot{u}}^i\left(t_j^{+}\right)-{\dot{u}}^i\left(t_j^{-}\right)=I_{ij}\left(u^i\left(t_j\right)\right),i=1,2,\cdots ,N;j=1,2,\cdots ,m.\end{array}$$ By using the least action principle and the saddle point theorem, some new existence theorems are obtained for second-order $p$-Laplacian systems with or without impulse under weak sublinear growth conditions, improving some existing results in the literature.

The existence of solutions for boundary value problems for a nonlinear discrete system involving the $p$-Laplacian is investigated. The approach is based on critical point theory.