This paper deals with the existence of solutions to the following system:$$\left\{\begin{array}{c}-\Delta u+u=\frac{\alpha }{\alpha +\beta }{a\left(x\right)\left|v\right|}^{\beta }{\left|u\right|}^{\alpha -2}u\text{in}{ℝ}^N\hfill \\ -\Delta v+v=\frac{\beta }{\alpha +\beta }{a\left(x\right)\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v\text{in}{ℝ}^N.\hfill \end{array}\right.$$−Δu+u=αα+βa(x)|v|β|u|α−2u inRN−Δv+v=βα+βa(x)|u|α|v|β−2v inRN. With the help of the Nehari manifold and the linking theorem, we prove the existence of at least two nontrivial solutions. One of them is positive. Our main tools are the concentration-compactness principle and the Ekeland’s variational principle.

We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation which nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem...

In this paper, we study the exterior boundary value problems of the Darwin model to the Maxwell’s equations. The variational formulation is established and the existence and uniqueness is proved. We use the infinite element method to solve the problem, only a small amount of computational work is needed. Numerical examples are given as well as a proof of convergence.

In this paper, we study the exterior boundary value problems of the Darwin model to the Maxwell's equations. The variational formulation is established and the existence and uniqueness is proved. We use the infinite element method to solve the problem, only a small amount of computational work is needed. Numerical examples are given as well as a proof of convergence.