具有临界增长的渐进周期拟线性Schrdinger方程的基态解(英文)

Ground states for asymptotically periodic quasilinear Schrdinger equations with critical growth

针对一类具有临界增长的渐进周期的拟线性Schrdinger方程,证明了基态解的存在性.首先利用一个变量代换,将拟线性Schrdinger方程转化为半线性Schrdinger方程.半线性Schrdinger方程的泛函在H1(RN)中定义良好,并且半线性Schrdinger方程和拟线性Schrdinger方程的基态解是一一对应的.然后利用山路引理证明了半线性Schrdinger方程的非平凡解的存在性.最后,在适当的单调性条件下,运用Nehari流形的方法和集中紧性引理证明了得到的非平凡解恰好是半线性Schrdinger方程的基态解.

For a class of asymptotically periodic quasilinear Schr？dinger equations with critical growth the existence of ground states is proved.First applying a change of variables the quasilinear Schr？dinger equations are reduced to semilinear Schr？dinger equations in which the corresponding functional is well defined in H1 RN .Moreover there is a one-to-one correspondence between ground states of the semilinear Schr？dinger equations and the quasilinear Schr？dinger equations.Then the mountain-pass theorem is used to find nontrivial solutions for the semilinear Schr？dinger equations. Finally under certain monotonicity conditions using the Nehari manifold method and the concentration compactness principle the nontrivial solutions are found to be exactly the same as the ground states of the semilinear Schr？dinger equations.