论文标题
在具有谐波潜力和联合非线性的情况下，总pitaevskii方程的许多积极解决方案
Infinitely many positive solutions of a Gross-Pitaevskii equation in the presence of a harmonic potential and combined nonlinearities
论文作者
论文摘要
本文的主要目的是在存在谐波电位的情况下解决微分方程领域的重要猜想。尽管在亚临界情况下，Hirose和Ohta在2007年已经解决了积极解决方案的独特性，但在超临界情况下，该问题一直持开放态度。在Hadj Selem等人中，作者获得了有趣的数值计算，这表明对于某些分叉参数$λ$，该方程具有许多在Infinity消失的正溶液。在本文中，我们通过构建责任数量的解决方案，这些解决方案是从独特的单数解决方案分叉的，$λ$接近Harmonic Operator $-Δ+ | x |^2 $。我们的方法取决于匹配的论点，并适用于超临界情况，以及在存在亚临界，批判性或超临界扰动的情况下的超批评案例。
The main goal of this paper is to address an important conjecture in the field of differential equations in the presence of a harmonic potential. While in the subcritical case, the uniqueness of positive solution has been addressed by Hirose and Ohta in 2007, the problem has remained open for years in the supercritical case. In Hadj Selem et al., the authors obtained interesting numerical computations suggesting that for some bifurcating parameter $λ$, the equation has many positive solutions that vanish at infinity. In this paper, we provide a proof to this claim by constructing an accountable number of solutions that bifurcate from the unique singular solutions with $λ$ close to the first eigenvalue $λ_1$ of the harmonic operator $-Δ+ |x|^2$. Our method hinges on a matching argument, and applies to the supercritical case, and to the supercritical case in the presence of a subcritical, critical or supercritical perturbation.
