A comprehensive study of nonlinear problems via variational approaches

我们通过变分方法研究非线性问题。特别是（1）研究了非线性Schrodinger方程和系统的奇摄动问题。我们引入了一个新的纯变分方法，使我们能够在一个非常一般的设置，以构建集中的解决方案。(2)我们研究非线性椭圆方程和系统在各种设置。给出了径向对称基态的一种新的变分构造。我们还研究了解的稳定性和不稳定性。(3)我们还研究了一维奇异摄动问题的高振荡解。我们给出了刻画和存在性结果。

We study nonlinear problems via variational approaches. Especially (1) we study singular perturbation problems for nonlinear Schrodinger equations and systems. We introduce a new purely variational method which enables us to construct concentrating solutions in a very general setting. (2) We study nonlinear elliptic equations and systems in various settings. We give a new variational construction of radially symmetric ground states. We also study stability and instability of solutions. (3) We also study highly oscillatory solutions in 1-dimensional singular perturbation problems. We give characterization and existence result.