Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies in three and four dimensions
Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies in three and four dimensions
复制标题
涉及三维和四维高频下 Sobolev 临界指数的组合功率型非线性标量场方程的基态唯一性
DOI：
10.1007/s00030-022-00804-0
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Murata Miho
中科院分区：
文献类型：
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作者：
Akahori Takafumi;Murata Miho
研究了在高频下含Sobolev临界指数的组合幂型非线性标量场方程基态的唯一性。在五个维度和更高维度中的高频率处的基态的唯一性已经在Akahori等人（Calc.变种偏微分方程58：32，2019）。此外，在三维中，可以从Coles和Gustafson的结果（Publ.Res.Inst.Math.Sci. 56：647-699，2020）。另一方面，在四个维度上的独特性还没有完全显现出来。本文的目的是统一地证明三维和四维中的唯一性。因此，我们得到了一个完整的答案，在高频率的基态的唯一性问题。从基态在无限频率下的极限轮廓（Aubin-Talenti函数）的角度来看，三维和四维的唯一性问题比高维的更困难。在本文中，我们采用了不动点的论点，在Coles和Gustafson（公共研究所数学科学。56：647-699，2020）。自从应用Coles和Gustafson的论点（Publ. Res. Inst. Math. Sci. 56：647-699，2020）到四维决不是简单的，我们需要构造一些估计的扰动预解式，适合不动点论点（见命题1.2和）。
We consider the uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies. The uniqueness of ground states at high frequencies in five and higher dimensions has been proved in Akahori et al. (Calc. Var. Partial Differential Equations 58:32, 2019). Moreover, that in three dimensions can be obtained from the result of Coles and Gustafson (Publ. Res. Inst. Math. Sci. 56:647–699, 2020). On the other hand, the uniqueness in four dimensions has not been completely revealed. The aim in this paper is to prove the uniqueness in three and four dimensions in a unified way. Thus, we obtain a complete answer to the uniqueness problem for ground states at high frequencies. From the point of view of limiting profile of ground states at infinite frequency which is known to be the Aubin–Talenti function, the uniqueness problem in three and four dimensions is more difficult than that in higher dimensions. In this paper, we employ the fixed-point argument developed in Coles and Gustafson (Publ. Res. Inst. Math. Sci. 56:647–699, 2020). Since the application of the argument of Coles and Gustafson (Publ. Res. Inst. Math. Sci. 56:647–699, 2020) to four dimensions is by no means straightforward, we need to construct some estimates for the perturbed resolvents which fit the fixed-point argument (see Proposition 1.2 and ).