Concentrated solutions to the Schr\"odinger--Bopp--Podolsky system with a positive potential

TL;DR

This paper studies the Schr"odinger--Bopp--Podolsky system with positive potential, proving the existence and multiplicity of solutions that concentrate near critical points of the potential as a small parameter tends to zero.

Contribution

It establishes new results on the existence and multiplicity of solutions to the Schr"odinger--Bopp--Podolsky system with positive potential, especially focusing on solutions concentrating around critical points.

Findings

Solutions concentrate around critical points of the potential

Multiplicity of solutions depends on the topology of the critical manifold

Existence results hold for small epsilon values

Abstract

Consider the Schr\"odinger--Bopp--Podolsky system \[ \begin{cases} -\epsilon^2\Delta u+(V+K\phi)u=u|u|^{p-1};\newline \Delta^2\phi-\Delta\phi=4\pi K u^2 \end{cases} ~\text{in}~\mathbb{R}^3 \] for sufficiently small $\epsilon>0$, where $V,K\colon\mathbb{R}^3\to [0,\infty[$; $p\in ]1,5[$ are fixed and we want to solve for $u,\phi\colon\mathbb{R}^3\to\mathbb{R}$. Under certain hypotheses, we estimate the multiplicity of solutions in function of a critical manifold of $V$ and we establish the existence of solutions concentrated around critical points of $V$.