ABSTRACT

In this paper, using a change of variables and variational method, positive solutions of the stationary relativistic nonlinear Schrödinger equation involving critical exponent and Hardy potential are studied when the potential function has positive lower bound and radial symmetry. We extend the result of Huang, Xiang (Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Commun Pure Appl Anal. 2016;15(4):1309–1333.) to equation with Hardy potential. But, it seems difficult to get solutions for our problem in $H^1\left({\mathbb{R}}^N\right)\cap L^{\mathrm{\infty }}\left({\mathbb{R}}^N\right)$ by perturbation approach as them since the existence of Hardy potential term.

摘要

本文采用变变量和变分法，研究了当势函数具有正下界和径向对称性时，涉及临界指数和Hardy势的稳态相对论非线性薛定谔方程的正解。我们将 Huang, Xiang（具有临界指数的拟线性薛定谔方程的孤子解。Commun Pure Appl Anal. 2016;15(4):1309–1333.）的结果扩展到具有哈代势的方程。但是，似乎很难为我们的问题找到解决方案$H^1\left({\mathbb{R}}^{ñ}\right)\cap {\mathrm{大号}}^{\mathrm{\infty }}\left({\mathbb{R}}^{ñ}\right)$由微扰方法作为它们自哈代势项的存在。