The new observations about the parameter-dependent Schrödinger-Poisson system

In this paper, we study the existence results of solutions for the following Schrödinger-Poisson system involving different potentials: {equation*} {cases} -Δu+V(x)u-λϕu=f(u)&~ R^3, -Δϕ=u^2&~ R^3. {cases} {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 {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.