Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies
Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies
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DOI：
10.1007/s00526-019-1556-6
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发表时间：
2018-01
影响因子：
2.1
通讯作者：
Takafumi Akahori;S. Ibrahim;N. Ikoma;Hiroaki Kikuchi;H. Nawa
中科院分区：
文献类型：
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作者：
Takafumi Akahori;S. Ibrahim;N. Ikoma;Hiroaki Kikuchi;H. Nawa
半线性椭圆方程基态解的唯一性和非退化性的研究是非常重要的，因为由此产生的能量景观及其对各种动力学的影响。Akahori等人（Global dynamics above the ground state energy for the combined power-type nonlinear Schrodinger equation with energy-critical growth at low frequencies，preprint）研究了具有包含Sobolev临界指数的组合幂型非线性的半线性椭圆方程。在那里，它表明，如果维度是4或更高，频率是足够小的，那么正径向基态是唯一的和非简并的。在本文中，我们将这些结果的情况下，高频率时的维数是5和更高。在对方程进行适当的重新标度后，我们证明了解的主要行为由索博列夫临界部分给出，其中基态是显式的，并且它们的简并性得到了很好的表征。我们的结果是一个关键的一步，对相应的非线性薛定谔和克莱因-戈登方程的能量以上的基态能量的解决方案的不同动力学的研究。我们对维度的限制主要是由于在第三和第四维度中存在共振。
The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In Akahori et al. (Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equation with energy-critical growth at low frequencies, preprint), semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schrödinger and Klein–Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.