Non-existence of ground states and gap of variational values for 3D Sobolev critical nonlinear scalar field equations
Non-existence of ground states and gap of variational values for 3D Sobolev critical nonlinear scalar field equations
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3D Sobolev 临界非线性标量场方程不存在基态和变分值间隙
DOI：
10.1016/j.jde.2022.06.016
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发表时间：
2022
影响因子：
2.4
通讯作者：
Nawa Hayato
中科院分区：
文献类型：
--
作者：
Akahori Takafumi;Ibrahim Slim;Kikuchi Hiroaki;Nawa Hayato
本文研究了三维空间中含Sobolev临界指数的组合幂型非线性标量场方程的极小化问题。在四个或更高的空间维度中，已知对于亚临界非线性的任何频率和任何功率，都存在基态。与这些情况相反，当空间维度为3且亚临界功率为3或更小时，我们可以证明存在一个阈值频率，高于该阈值频率不存在基态，低于该阈值频率存在基态(见定理1.1和1.2)。此外，我们还证明了用于刻画基态的两个典型变分问题之间的区别(见定理1.3)。
In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists (see Theorems 1.1 and 1.2). Furthermore, we prove the difference between two typical variational problems used to characterize the ground states (see Theorem 1.3).