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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.12732 |
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Table of Contents:
- In this paper, we study the existence results of solutions for the following Schrödinger-Poisson system involving different potentials: \begin{equation*} \begin{cases} -Δu+V(x)u-λϕu=f(u)&\quad\text{in}~\mathbb R^3, -Δϕ=u^2&\quad\text{in}~\mathbb R^3. \end{cases} \end{equation*} We first consider the case that the potential $V$ is positive and radial so that the mountain pass theorem could be implied. The other case is that the potential $V$ is coercive and sign-changing, which means that the Schrödinger operator $-Δ+V$ 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 $f$ 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 Schrödinger-Poisson system.