| Abstrakt: |
In this paper, we investigate normalized solutions for the following Schrödinger-Poisson system with an $ L^2 $-constraint:$ \begin{equation*} \label{SP1} \left\{ \begin{array}{ll} -\Delta u+\lambda u+|x|^2u+\frac{1}{2\pi}\left(\ln|\cdot|\ast|u|^2\right)u = f(u), & x\in \mathbb{R}^2, \\ \int_{ \mathbb{R}^2}u^2\mathrm{d}x = c, \ \end{array} \right. \end{equation*} $where $ f\in \mathcal{C}(\mathbb{R}, \mathbb{R}) $, $ c>0 $ is a given real number and $ \lambda\in \mathbb{R} $ arises as a Lagrange multiplier. By developing some new mathematical strategies and analytical techniques, we overcome the difficulties posed by the logarithmic convolution term $ \ln|\cdot|\ast|u|^2 $ and the trapping potential $ |x|^2 $, and prove the existence of normalized solutions to the above system under some suitable assumptions on $ f $. Notably, we extend the existing results concerning $ L^2 $-subcritical, $ L^2 $-critical, $ L^2 $-supercritical cases with $ c\in(0, c_0) $ to arbitrary $ c>0 $. Our method not only extends the existence range of mass solutions in the pure $ L^2 $-supercritical case, but also introduces a novel approach to investigating nonhomogeneous $ L^2 $-constrained problems with mixed nonlinearities. [ABSTRACT FROM AUTHOR] |
