论文标题
在X射线无电子激光schrödinger方程的站立波上
On the standing waves for the X-ray free electron laser Schrödinger equation
论文作者
论文摘要
在本文中，我们关注以下非线性schrödinger方程$$ i \ partial_ {t}ψ=-Δψ+b^2（x_1^2+x_2^2）ψ+\ frac {λ_1} λ_3|ψ|^pψ，~~~（t，x）\ in \ mathbb {r}^+\ times \ times \ mathbb {r}^3，$ 0，其中$ 0 <p <4 $。我们主要研究了该方程式的常规波的存在和稳定性/不稳定性，在两种情况下：第一种情况是不涉及磁性电位（即等式中的$ b = 0 $），第二个是$ b \ b \ neq 0 $。确切地说，在第一种情况下，通过考虑在合适的Pohozaev歧管上的最小化问题，我们证明了基态的存在，并进一步证明了所有基态立场在有限的时间内都被爆炸而言是强烈的不稳定。此外，通过利用其证明的思想，我们能够证明与现有文献中归一化解决方案的研究相比，标准化解决方案的存在和不稳定性似乎是新的。在第二种情况下，由于部分谐波潜力的额外术语，情况更难治疗。我们设法证明了（0,4）$ in（0,4）$的稳定站立波的存在，并在系数上进行了一些假设，如果$ p \ in（0，\ frac {4} {3} {3}] $，则作为全局最小化的解决方案作为全局最小化，以及本地最小化的$ p \ in [$ p \ in [$ p \ in [\ frac frac} $ p \ frac};在[\ frac {4} {3}，4）$中的质量关键和超临界情况下，我们在合适的歧管上建立了基态的变异表征，这与nehari类型或pohari类型都不一样，然后证明地面状态的存在。最终，在$ω$和$ p $的一些假设下，我们证明地面立场强烈不稳定。
In this paper, we are concerned with the standing waves for the following nonlinear Schrödinger equation $$i\partial_{t}ψ=-Δψ+b^2(x_1^2+x_2^2)ψ+\frac{λ_1}{|x|}ψ+ λ_2(|\cdot|^{-1}\ast |ψ|^2)ψ- λ_3|ψ|^p ψ,~~~ (t,x)\in \mathbb{R}^+\times \mathbb{R}^3,$$ where $0<p<4$. We mainly study the existence and stability/instability properties of standing waves for this equation, in two cases: the first one is that no magnetic potential is involved, (i.e. $b=0$ in the equation) and the second one is that $b\neq 0$. To be precise, in the first case, by considering a minimization problem on a suitable Pohozaev manifold we prove the existence of ground states, and show further that all ground state standing waves are strongly unstable by blow-up in finite time. Moreover, by making use of the ideas of their proofs, we are able to prove the existence and instability of normalized solutions, whose proofs seem to be new, compared with the studies of normalized solutions in the existing literature. In the second case, the situation is more difficult to be treated, due to the additional term of the partial harmonic potential. We manage to prove the existence of stable standing waves for $p\in (0,4)$ and with some assumptions on the coefficients, where solutions are obtained as global minimizers if $p\in (0,\frac{4}{3}]$, and as local minimizers if $p\in [\frac{4}{3}, 4)$. In the mass-critical and supercritical cases $p\in [\frac{4}{3}, 4)$, we establish the variational characterization of the ground states on a suitable manifold which is different from the one neither of the Nehari type nor of the Pohozaev type, and then prove the existence of ground states. Finally under some assumptions on $ω$ and $p$, we prove that the ground state standing waves are strongly unstable.
