The new observations about the parameter-dependent Schr\"{o}dinger-Poisson system
Abstract
In this paper, we study the existence results of solutions for the following Schr\"{o}dinger-Poisson system involving different potentials: \begin{equation*} \begin{cases} -\Delta u+V(x)u-\lambda \phi u=f(u)&\quad\text{in}~\mathbb R^3, -\Delta\phi=u^2&\quad\text{in}~\mathbb R^3. \end{cases} \end{equation*} We first consider the case that the potential is positive and radial so that the mountain pass theorem could be implied. The other case is that the potential is coercive and sign-changing, which means that the Schr\"{o}dinger operator is allowed to be indefinite. To deal with this more difficult case, by a local linking argument and Morse theory, the system has a nontrivial solution. Furthermore, we also show the asymptotical behavior result of this solution. Additionally, the proofs rely on new observations regarding the solutions of the Poisson equation. As a main novelty with respect to corresponding results in \cite{MR4527586,MR3148130,MR2810583}, we only assume that satisfies the super-linear growth condition at the origin. We believe that the methodology developed here can be adapted to study related problems concerning the existence of solutions for Schr\"{o}dinger-Poisson system.
Cite
@article{arxiv.2508.12732,
title = {The new observations about the parameter-dependent Schr\"{o}dinger-Poisson system},
author = {Chen Huang and Sihua Liang and Lei Ma and Patrizia Pucci},
journal= {arXiv preprint arXiv:2508.12732},
year = {2026}
}