Multiplicity and concentration results for fractional Schr\"odinger-Poisson equations with magnetic fields and critical growth

TL;DR

This paper investigates the existence, multiplicity, and concentration behavior of solutions to a fractional Schr"odinger-Poisson equation with magnetic fields and critical growth, as the parameter psilon approaches zero, linking solutions to the topology of the potential's minimum set.

Contribution

It establishes new results on the multiplicity and concentration of solutions for fractional Schr"odinger-Poisson equations with magnetic fields, considering critical growth and linking solutions to the topology of the potential's minimum.

Findings

Multiple solutions concentrate near the minimum points of the potential.

The number of solutions relates to the topological complexity of the potential's minimum set.

Solutions exhibit concentration phenomena as psilon .

Abstract

We deal with the following fractional Schr\"odinger-Poisson equation with magnetic field \begin{equation} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{2^{*}_{s}-2}u \quad \mbox{in} \mathbb{R}^{3}, \nonumber \end{equation} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $t\in (0,1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a subcritical nonlinearity. Under a local condition on the potential $V$, we study multiplicity and concentration of nontrivial solutions as $\varepsilon\rightarrow 0$. In particular, we relate the number of nontrivial solutions with the topology of the set where the potential $V$ attains its minimum.