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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.16368 |
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Table of Contents:
- We investigate the existence, uniqueness, and radial symmetry of normalized solutions to the Schrödinger Poisson equation with non-autonomous nonlinearity $f(x,u)$: \begin{equation} -\triangle u+(|x|^{-1}*|u|^2)u=f(x,u)+λu, \nonumber \end{equation} subject to the constraint $\mathcal{S}_c=\{u\in H^1(\mathbb{R}^3)|\int_{\mathbb{R}^3}u^2=c>0 \}$. We consider three cases based on the behavior of $f(x,u)$: the $L^2$ supercritical case, the $L^2$ subcritical case with growth speed less than three power times, and the $L^2$ subcritical case with growth speed more than three power times. We establish the existence of solutions using three different methods depending on $f(x,u)$. Furthermore, we demonstrate the uniqueness and radial symmetry of normalized solutions using an implicit function framework when $c$ is small.