一类无紧性扰动拟线性薛定谔方程的解

Solutions to a Class of Disturbed Quasilinear Schrodinger Equations without Compactness

利用Nehari流形方法研究了一类带有扰动项的拟线性薛定谔方程基态解的存在性。首先,利用一个代数方程证明了方程对应的Nehari流形是非空的。其次,根据流形的定义以及Sobolev不等式,证明了当限制在Nehari流形时元素范数有正下界。然后,利用集中紧性原理解决了工作空间紧性缺失的问题,进而得到方程对应泛函限制极小值的可达性。最后,利用条件极值原理得到方程基态解的存在性。

The existence of ground state solutions for a class of quasilinear Schr?dinger equations with disturbance term was studied through Nehari manifold method.Firstly,it was proved that the Nehari manifold corresponding to the equation was non-empty by using an algebraic equation.Secondly,according to the definition of manifold and the Sobolev inequality,the norm of the elements had a positive lower bound when the Nehari manifold was limited.And then,the lack problem of compactness in the working space was solved by the concentration compactness principle.Thus,the functional constraint minimum corresponding to the equation was obtained.Finally,the existence of the ground state solution of the equation was obtained by using the constrained extremum principle.