一类非线性分数阶薛定谔-泊松系统非零解的存在性

Existence of Nontrivial Solutions for a Class of Nonlinear Fractional Schr\"{o}dinger-Poisson System

中文关键词:  分数阶薛定谔-泊松系统  非平凡解  扰动法；Morse迭代法

英文关键词:fractional Schr\"{o}dinger-Poisson system  nontrivial solution  perturbation method  Moser iterative method

中文摘要:

      本文研究了分数阶薛定谔-泊松系统$$\left\{\begin{array}{l}(-\Delta)^su+u+\phi u=\lambda f(u)\ \text {in} \ \mathbb {R}^3, \\ (-\Delta)^{\alpha}\phi =u^2\ \text {in} \ \mathbb {R}^3\emph{},\end{array}\right. $$ 非零解的存在性, 其中$s\in (\frac{3}{4},1), \alpha\in(0,1),\lambda$ 是正参数, $(-\Delta)^s,(-\Delta)^{\alpha}$是分数阶拉普拉斯算子. 在一定的假设条件下, 利用扰动法和Morse迭代法, 得到了系统至少一个非平凡解.

英文摘要:

      {R}^3\emph{},\end{array}\right.$$ where $s\in (\frac{3}{4},1), \alpha\in(0,1),\lambda$ is a positive parameter, $(-\Delta)^s,(-\Delta)^{\alpha}$ are fractional Laplacian operators. Under certain assumptions on $f$, we obtain the existence of at least one nontrivial solution of the system by using the methods of perturbation and Moser iterative method.