Schrödinger-Bopp-Podolsky system with sublinear and critical nonlinearities: solutions at negative energy levels and asymptotic behaviour

Abstract

We consider the following Schrödinger-Bopp-Podolsky system with critical and sublinear terms \begin{equation*} \begin{cases} - Δu+ u+Q(x)ϕu= \vert u\vert^4 u+ λK(x)\vert u \vert^{p-1}u&\mbox{ in }\ \mathbb{R}^3 \smallskip - Δϕ+ a^{2}Δ^{2} ϕ= 4πQ(x) u^{2}& \mbox{ in }\ \mathbb{R}^3. \end{cases} \end{equation*} Here $u,\ddot{I}•:{\mathbb{R}}^3\to \mathbb{R}$ are the unknowns, $Q$ and $K$ are given functions satisfying mild assumptions, $a\ge 0,\hat{I}»>0$ are parameters and $p\in (0,1)$. We first show existence of infinitely many solutions at negative energy level, including the ground state, when the parameter $\hat{I}»$ is small. Then we give general results concerning the structure of the set of solutions. We show also the behaviour of the solutions as the parameters $a,\hat{I}»$ tend to zero. In particular the ground states solutions tends to a ground state solution of the Schrödinger-Poisson system as $a$ tends to zero.