2025/08/18 by Chen Huang, Sihua Liang, Huang, Chen +4
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2508.12732
openalex publication_date 2025/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the existence results of solutions for the following Schrödinger-Poisson system involving different potentials: \begincases -Δu+V(x)u-λϕu=f(u)amp;\quadin~\mathbb R3, -Δϕ=u2amp;\quadin~\mathbb R3. \endcases 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 \citeMR4527586,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.