Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.
The finite element analysis of linear elliptic problems in two-dimensional domains with cusp points (turning points) is presented. This analysis needs on one side a generalization of results concerning the existence and uniqueness of the solution of a constinuous elliptic variational problem in a domain the boundary of which is Lipschitz continuous and on the other side a presentation of a new finite element interpolation theorem and other new devices.
We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension . We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if ).