We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the $GL$-energy $E_{\epsilon }$ and the parameter $\epsilon$. These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.

In this paper we investigate the existence of solutions to impulsive problems with a $p\left(t\right)$-Laplacian and Dirichlet boundary value conditions. We introduce two types of solutions, namely a weak and a classical one which coincide because of the fundamental lemma of the calculus of variations. Firstly we investigate the existence of solution to the linear problem, i.e. a problem with a fixed rigth hand side. Then we use a direct variational method and next a mountain pass approach in order to get the existence...

Using Ricceri's variational principle, we establish the existence of infinitely many solutions for a class of two-point boundary value Kirchhoff-type systems.