Normalized solution for fractional Choquard equation with potential and general nonlinearity

Abstract

In this paper, we consider the following fractional Choquard equation: $\{\begin{array}{l}(-\mathrm{\Delta }{)}^{s}u+V(x)u+\mathrm{\lambda u}=(I_{\alpha }\ast F(u))f(u)\mathrm{in}{\mathbb{R}}^N,\\ {\displaystyle {\int }_{{\mathbb{R}}^N}{u}^2\mathrm{d}x={\rho }^2,\rho >0,}\end{array}$ where $s\in (0,1)$, $\alpha \in (0,N)$, N>2s and $V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous nonnegative potential and vanishes at infinity. Under some mild assumptions imposed on V and f, we establish the existence of $L^2$-normalized solution $(u,\lambda )\in H^s({\mathbb{R}}^N)\times {\mathbb{R}}^{+}$.

Keywords:

• Normalized solution
• fractional Choquard equation
• potential
• stretched functional

AMS Subject Classifications:

• 35A15
• 35J60
• 58E05

Disclosure statement

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

Additional information

Funding

H. R. Sun was partly supported by the NSFC (Grant No. 11671181) NSF of Gansu Province of China (Grant No. 21JR7RA535) and Gansu Provincial Department of Education: young doctor fund project (2022QB-001). J. J. Zhang was supported by NSFC (Grant No. 11871123).