论文标题
非线性Schrödinger-Newton系统的新型解决方案
New type of solutions for the nonlinear Schrödinger-Newton system
论文作者
论文摘要
非线性Schrödinger-Newton System \ begin {等式*} \ begin {case} Δu-v（| x |）u +ψu= 0，＆〜x \ in \ mathbb {r}^3，\\Δψ + \ frac12 u^2 = 0 = 0，＆〜x \ in \ m athbb {r}^3，\ end e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e n in line con e ny cout in line cou。带有牛顿力学重力法的量子力学。 WEI和YAN（CALC。VAR。部分微分方程37（2010），423--439）证明，Schrödinger方程在$ \ Mathbb {r}^n $中具有无限的许多积极解决方案，并且这些解决方案在$（y_ _ {1}，y _ _ {2} 2} 2} $ insection中，它们具有多边形对称性。 Duan等。在（Arxiv：2006.16125v1）扩展了WEI和YAN获得的结果，这些解决方案在$（y_ {1}，y_ {2}）$平面中具有多边形对称性，甚至在$ y_ {2} $中，在$ y_ {2} $中，还有一个在selutions.hu等人表达的参数。在对潜在函数V的适当假设下，Hu等人。在（ARXIV：2106.04288V1）中，为Schrödinger-Newton系统构建了许多非主体阳性解决方案，这些阳性解决方案在$（y_ {1}，y_ {2}）中具有多边形对称性，它们甚至是$ y_ __ {2} $ y_ {2} $ y_ y_ y_ y_ y_ y_ y_ y_ y_ _ _} $ y_ __ p}。假设$ v（r）$具有以下字符\ begin {qore*} v（r）= v_ {1}+\ frac {b} {r^q}+o \ big（\ frac {\ frac {1} {r^{q+σ}}}}}}}}} \ big） $ \ frac12 \ leq q <1 $和$ b，v_ {1}，σ$是一些正常数，$ v（y）\ geq v_1> 0 $，我们构建了无限的许多非范围的正面解决方案，这些解决方案在$（y__ {1}，y__ {2} $ y__ {2} $ y____________________________2中Schrödinger-Newton系统由Lyapunov-Schmidt减少方法。我们扩展了Duan等人获得的结果。在（Arxiv：2006.16125v1）中，到非线性Schrödinger-Newton系统。
The nonlinear Schrödinger-Newton system \begin{equation*} \begin{cases} Δu- V(|x|)u + Ψu=0, &~x\in\mathbb{R}^3,\\ ΔΨ+\frac12 u^2=0, &~x\in\mathbb{R}^3, \end{cases} \end{equation*} is a nonlinear system obtained by coupling the linear Schrödinger equation of quantum mechanics with the gravitation law of Newtonian mechanics. Wei and Yan in (Calc. Var. Partial Differential Equations 37 (2010),423--439) proved that the Schrödinger equation has infinitely many positive solutions in $\mathbb{R}^N$ and these solutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they are radially symmetric in the other variables. Duan et al. in (arXiv:2006.16125v1) extended the results got by Wei and Yan and these solutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they are even in $y_{2}$with one more more parameter in the expression of the solutions.Hu et al. Under the appropriate assumption on the potential function V, Hu et al. in (arXiv: 2106.04288v1) constructed infinitely many non-radial positive solutions for the Schrödinger-Newton system and these positive solutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they are even in $y_{2}$ and $y_{3}$. Assuming that $V(r)$ has the following character \begin{equation*} V(r)=V_{1}+\frac{b}{r^q}+O\Big(\frac{1}{r^{q+σ}}\Big),~\mbox{ as } r\rightarrow\infty, \end{equation*} Where $\frac12\leq q<1$ and $b, V_{1}, σ$ are some positive constants, $V(y)\geq V_1>0$, we construct infinitely many non-radial positive solutions which have polygonal symmetry in the $(y_{1}, y_{2})$ plane and are even in $y_{2}$ for the Schrödinger-Newton system by the Lyapunov-Schmidt reduction method. We extend the results got by Duan et al. in (arXiv:2006.16125v1) to the nonlinear Schrödinger-Newton system.
