Schrodinger-Kirchhoff equation involving double critical nonlinearities
Schrodinger-Kirchhoff equation involving double critical nonlinearities
复制标题
涉及双临界非线性的薛定谔-基尔霍夫方程
DOI：
10.1016/j.jmaa.2018.10.079
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发表时间：
2019
影响因子：
1.3
通讯作者：
Zhang Xinguang
中科院分区：
文献类型：
--
作者：
Guo Zuji;Zhang Xinguang
本文研究了具有双临界非线性项的Schrödinger-Kirchhoff方程{−（a+ B）R 3| u| 2）Δ u= u 2 <$（s 1）− 1| X| s 1+ u 2 <$（s 2）− 1| X| s 2，在R3中（s）= 2（3− s）且0≤ s 1< s 2< 2。一个自然的策略是通过变分方法寻找方程的弱解作为合适泛函的临界点。但通过伸缩不变性我们看到该泛函不满足（PS）条件。我们将使用几种技术来恢复紧凑性。结果表明，高阶项占主导地位的低阶项，并存在多尺度特性，在这个非线性问题。
In this paper, we are interested in weak solutions of the following Schrödinger–Kirchhoff equation involving double critical nonlinearities:{−(a+ b∫ R 3|∇ u| 2) Δ u= u 2⁎(s 1)− 1| x| s 1+ u 2⁎(s 2)− 1| x| s 2, in R 3∖{0}, u∈ D 1, 2 (R 3), u≥ 0 in R 3, where a, b are two positive constants, 2⁎(s)= 2 (3− s) and 0≤ s 1< s 2< 2. A natural strategy is to search the weak solutions of the equation as critical points of a suitable functional via variational methods. But by dilation invariance we see that the functional doesn't satisfy (P S) condition. We will make use of several techniques to recover compactness. Our results show that the higher order term dominates lower order term and there exist multi-scale characteristics in this nonlinear problem.