Vortex-type solutions for magnetic pseudo-relativistic Hartree equation

Abstract

We deal with a class of magnetic pseudo-relativistic Hartree type equation $\sqrt{(-i\mathrm{\nabla }-A(x){)}^2+m^2}u+W\left(x\right)u=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u,x\in {\mathbb{R}}^N,$ where $N\ge 3$, m>0, $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is a continuous vector potential, $W:{\mathbb{R}}^N\to \mathbb{R}$ is an external continuous scalar potential and $I_{\alpha }\left(x\right)=\left(c_{N,\alpha }/|x{|}^{N-\alpha }\right)(x\ne 0)$ is a convolution kernel, $c_{N,\alpha }>0$ is a positive constant, $2\le p<2N/\left(N-1\right)$, $(N-1)p-N<\alpha <N$. Under the action of some subgroup of linear isometries on potential A and W, and some assumptions on the decay of A and W at infinity, we prove the existence of vortex-type solutions to this problem by using variational methods and asymptotic estimates as p = 2.

Keywords:

• Pseudo-relativistic Hartree equation
• vortex-type solutions
• asymptotic estimates

Mathematics Subject Classifications (2010):

• 35Q40
• 35Q55

Disclosure statement

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

Additional information

Funding

This paper is supported by the National Natural Science Foundation of China (NNSF of China) [No. 11771291].