On the existence of normalized solutions to a class of fractional Choquard equation with potentials

Abstract

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$(-\Delta {)}^{s}u+V(x)u=\lambda u+f(x)\left(I_{\alpha }\ast \left(f|u{|}^q\right)\right)|u{|}^{q-2}u+g(x)\left(I_{\alpha }\ast \left(g|u{|}^p\right)\right)|u{|}^{p-2}u,x\in {\mathbb{R}}^N$$ subject to the mass constraint $${\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=a>0,$$ where $N>2s,s\in (0,1),\alpha \in (0,N)$, and $\frac{N+\alpha }{N}\le q<p\le \frac{N+\alpha +2s}{N}$. Here, the parameter $\lambda \in \mathbb{R}$ appears as an unknown Lagrange multiplier associated with the normalization condition. By employing variational methods under appropriate assumptions on the potentials $V(x),f(x)$, and $g(x)$, we establish several existence results for normalized solutions.

To appear in Acta Mathematica Scientia