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

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 $V$.