Ground states solutions for a modified generalized Choquard fractional Schrödinger equation

ABSTRACT

In this paper, we consider the modified fractional Schrödinger system of Choquard type: $\{\begin{array}{l}{(-\mathrm{△})}^{s}u+V_1(x)u+\kappa \left[{(-\mathrm{△})}^s{u}^2\right]\\ u=(|x{|}^{-\lambda }\ast F(u))f(u)+(I_{\lambda }\ast G(v))g(u)& \text{in}{\mathbb{R}}^{\mathbb{N}},\\ {(-\mathrm{△})}^{s}v+V_2(x)v+\kappa \left[{(-\mathrm{△})}^s{v}^2\right]\\ v=((|x{|}^{-\lambda }\ast F(v))f(v)+(I_{\lambda }\ast G(u))g(v)& \text{in}{\mathbb{R}}^{\mathbb{N}},\end{array}$ where ${(-\mathrm{△})}^{s}u$ is the fractional Laplacian with $s\in (0,1)$, $V_1(x),V_2(x)$ are nonnegative potentials, $\lambda ,\kappa$ are positive constant and $G(u)={\int }_0^{u}g(t)\mathrm{d}t$, $F(u)={\int }_0^{u}f(t)\mathrm{d}t$. Under some assumptions on $f,g,F$ and G, by using the variational method, we obtain ground state solutions to the system.

KEYWORDS:

• Fractional Schrödinger equation
• Choquard type equation
• ground states solution

AMS SUBJECT CLASSIFICATIONS:

• 35A15
• 35R11
• 35J60

Acknowledgments

The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.