Existence and nonexistence of positive solutions for a static Schrodinger-Poisson-Slater equation
Existence and nonexistence of positive solutions for a static Schrodinger-Poisson-Slater equation
复制标题
静态薛定谔-泊松-斯莱特方程正解的存在性和不存在性
DOI：
10.1016/j.jde.2018.10.048
复制
发表时间：
2019
影响因子：
2.4
通讯作者：
Huang Shuibo
中科院分区：
文献类型：
--
作者：
Liu Zhisu;Zhang Zhitao;Huang Shuibo
本文研究了一类具有临界增长的Schrödinger-Poisson-斯莱特方程-△ u+（u2 1| 4 π x|）u= μ| u| p− 1 u+| u| 4 u，其中μ> 0且p∈（11/7，5）.对于p∈（2，5）的情形。我们开发了一种新的微扰方法，与著名的山通定理，证明存在的正基态。对于p= 2的情形，通过限制μ的取值范围得到了非平凡解的不存在性，并利用约束极小化方法研究了正解的存在性.对于p∈（11/7，2）的情形，利用Brezis和Oswald [9]的截断技巧和Lions [27]的测度表示集中紧性原理，证明了当μ∈（0，μ ∈）> 0时径向对称正解的存在性.上述结果将Ianni和Ruiz [18]关于次临界情形的一些定理推广到了临界情形。
In this paper we study the following type of the Schrödinger–Poisson–Slater equation with critical growth−△ u+(u 2⋆ 1| 4 π x|) u= μ| u| p− 1 u+| u| 4 u, in R 3, where μ> 0 and p∈(11/7, 5). For the case of p∈(2, 5). We develop a novel perturbation approach, together with the well-known Mountion–Pass theorem, to prove the existence of positive ground states. For the case of p= 2, we obtain the nonexistence of nontrivial solutions by restricting the range of μ and also study the existence of positive solutions by the constrained minimization method. For the case of p∈(11/7, 2), we use a truncation technique developed by Brezis and Oswald [9] together with a measure representation concentration-compactness principle due to Lions [27] to prove the existence of radial symmetrical positive solutions for μ∈(0, μ⁎) with some μ⁎> 0. The above results nontrivially extend some theorems on the subcritical case obtained by Ianni and Ruiz [18] to the critical case.