Multiplicity and concentration results for fractional Kirchhoff equation with magnetic field

ABSTRACT

In the paper, we considered the fractional Kirchhoff equation with magnetic field $\begin{array}{r}(a{\epsilon }^{2s}+b{\epsilon }^{4s-3}[u{]}_{A/\epsilon }^2)(-\mathrm{\Delta }{)}_{A/\epsilon }^{s}u+V(x)u=f(|u{|}^2)u\text{in }{\mathbb{R}}^3,\end{array}$ where ε is a small parameter, a, b are positive constants, $s\in (\frac{3}{4},1)$, $A:{\mathbb{R}}^3\to {\mathbb{R}}^3$ is a Hölder continuous magnetic potential with exponent $\alpha \in (0,1]$, $[u{]}_{A/\epsilon }^2={\iint }_{{\mathbb{R}}^6}\frac{|u(x)-{\epsilon }^{ı(x-y)\cdot \frac{A}{\epsilon }(\frac{x+y}{2})}u(y){|}^2}{|x-y{|}^{3+2s}}\mathrm{d}x\mathrm{d}y$. With proper assumptions on V and f, we develop a new approach to prove the Palais–Smale condition for the corresponding energy functional and establish the multiplicity and concentration of solutions by using perturbation techniques and Ljusternik–Schnirelmann theory.

KEYWORDS:

• Fractional Kirchhoff equation
• magnetic field
• concentrating phenomenon
• Ljusternik–Schnirelmann theory

AMS SUBJECT CLASSIFICATIONS:

• 35D30
• 35A15
• 47G20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second author is supported by the National Natural Science Foundation of China [grant number 11471147].