Schr\"odinger-Bopp-Podolsky system with sublinear and critical nonlinearities: solutions at negative energy levels and asymptotic behaviour

TL;DR

This paper studies solutions to a Schr"odinger-Bopp-Podolsky system with critical and sublinear nonlinearities, proving existence of multiple solutions at negative energy levels and analyzing their asymptotic behavior as parameters vary.

Contribution

It establishes the existence of infinitely many solutions at negative energy levels for small parameters and describes the solutions' asymptotic behavior as parameters tend to zero.

Findings

Existence of infinitely many solutions at negative energy levels.

Ground state solutions converge to Schr"odinger-Poisson ground states as parameters tend to zero.

Solutions exhibit specific asymptotic behavior with parameter variations.

Abstract

We consider the following Schr\"odinger-Bopp-Podolsky system with critical and sublinear terms \begin{equation*} \begin{cases} - \Delta u+ u+Q(x)\phi u= \vert u\vert^4 u+ \lambda K(x)\vert u \vert^{p-1}u&\mbox{ in }\ \mathbb{R}^3 \smallskip - \Delta \phi+ a^{2}\Delta^{2} \phi = 4\pi Q(x) u^{2}& \mbox{ in }\ \mathbb{R}^3. \end{cases} \end{equation*} Here $u,\phi:\mathbb {R}^{3}\rightarrow \mathbb{R}$ are the unknowns, $Q$ and $K$ are given functions satisfying mild assumptions, $a\geq0, \lambda>0$ 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 $\lambda$ 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 $a,\lambda$ tend to zero. In particular the ground states solutions tends to a ground state solution of the Schr\"odinger-Poisson system as $a$ tends to zero.