论文标题
Schrodinger-Newton系统的归一化峰解决方案的存在和局部唯一性
Existence and local uniqueness of normalized peak solutions for a Schrodinger-Newton system
论文作者
论文摘要
在本文中，我们研究了诱捕势是退化且具有非分离的临界点的假设，我们研究了Schrödinger-Newton系统的归一化峰溶液的存在和局部唯一性。 首先，我们研究了Schrödinger-Newton系统的标准化单峰解决方案的存在和局部唯一性。确切地说，我们对化学势$μ$和有吸引力的交互作用$ a $进行精确描述。然后，我们应用有限的尺寸还原方法来获得单峰溶液的存在。此外，使用各种局部Pohozaev身份，爆破分析和最大原则，我们通过精确分析集中点和Lagrange乘数来证明单峰溶液的局部唯一性。最后，我们还证明了Schrödinger-Newton系统的多峰解决方案不存在，该解决方案与相应的Schrödinger方程式明显不同。非本地术语导致这种差异。 主要困难来自Lagrange乘数的估计值，即$ P（X）$的临界点的不同方向的不同退化率以及非本地术语涉及的一些复杂估计。据我们所知，这可能是第一次使用规定的$ l^{2} $ - Schrödinger-Newton System的规定的解决方案的存在和局部唯一性。
In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schrödinger-Newton system. Precisely, we give the precise description of the chemical potential $μ$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local Pohozaev identities, blow-up analysis and the maximum principle, we prove the local uniqueness of single-peak solutions by precise analysis of the concentrated points and the Lagrange multiplier. Finally, we also prove the nonexistence of multi-peak solutions for the Schrödinger-Newton system, which is markedly different from the corresponding Schrödinger equation. The nonlocal term results in this difference. The main difficulties come from the estimates on Lagrange multiplier, the different degenerate rates along different directions at the critical point of $P(x)$ and some complicated estimates involved by the nonlocal term. To our best knowledge, it may be the first time to study the existence and local uniqueness of solutions with prescribed $L^{2}$-norm for the Schrödinger-Newton system.
