On the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

TL;DR

This paper studies fractional Schr"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearities, proving existence and multiplicity of solutions using variational methods and concentration compactness principles.

Contribution

It introduces new existence and multiplicity results for fractional Schr"{o}dinger-Kirchhoff equations with electromagnetic fields under critical nonlinearities.

Findings

Existence of at least one solution for small epsilon.

Multiple solutions exist depending on the parameter epsilon.

Solutions tend to zero as epsilon approaches zero.

Abstract

We consider the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity $\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-\Delta)_{A_\varepsilon}^su + V(x)u =$ $|u|^{2_s^\ast-2}u + h(x,|u|^2)u,$ $\ \ x\in \mathbb{R}^N,$ where $ u(x) \rightarrow 0$ as $|x| \rightarrow \infty,$ and $(-\Delta)_{A_\varepsilon}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s),$ $M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}$ is a continuous nondecreasing function, $V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0,$ and $A: \mathbb{R}^N \rightarrow \mathbb{R}^N$ are the electric and the magnetic potential, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that $\varepsilon < \mathcal {E}$; and (ii) for any $m^\ast \in \mathbb{N}$, has $m^\ast$ pairs of solutions if $\varepsilon < \mathcal {E}_{m^\ast}$, where $\mathcal {E}$ and $\mathcal {E}_{m^\ast}$ are sufficiently small positive numbers. Moreover, these solutions $u_\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$.