论文标题
在球外部的NLS中，具有纯正功能非线性的NLS的正质量的存在，不存在和多重溶液的阈值
Threshold for Existence, Non-existence and Multiplicity of positive solutions with prescribed mass for an NLS with a pure power nonlinearity in the exterior of a ball
论文作者
论文摘要
我们获得了球外部的半线性椭圆方程的归一化解决方案的存在，不存在和多样化的阈值结果。据我们所知，这是解决这个问题的文献中的第一个结果。特别是，我们表明规定的质量会影响归一化解决方案的数量，并且在质量超临界情况下具有稳定作用。此外，在阈值中，当n = 2时，我们发现一个新的指数p = 6，过去似乎并没有为此方程发挥作用。此外，我们的发现“非常令人惊讶”，与在整个空间和球上获得的结果完全不同。我们还将证明该域的性质对于站立波的存在和稳定性至关重要。作为预言，众所周知，在超临界情况下，这些波在RN中是不稳定的。在本文中，我们将证明它们在外部领域非常稳定。
We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surprising" and completely different from the results obtained on the entire space and on balls. We will also show that the nature of the domain is crucial for the existence and stability of standing waves. As a foretaste, it is well-known that in the supercritical case these waves are unstable in RN . In this paper, we will show that in the exterior domain they are strongly stable.
