Normalized Solutions for the Fractional Choquard Equations with Lower Critical Exponent and Nonlocal Perturbation

Abstract

We study the existence and nonexistence of normalized solutions for the following fractional Choquard equations with Hardy–Littlewood–Sobolev lower critical exponent and nonlocal perturbation: \[ \begin{cases} (-\Delta)^{s} u + \lambda u = \gamma (I_{\alpha} \ast |u|^{\frac{\alpha}{N}+1}) |u|^{\frac{\alpha}{N}-1} u + \mu (I_{\alpha} \ast |u|^{q}) |u|^{q-2} u &\textrm{in $\mathbb{R}^{N}$}, \\ \int_{\mathbb{R}^{N}} |u|^{2} \, \mathrm{d}x = c^{2}, \end{cases} \] where $N \geq 3$, $s \in (0,1)$, $\alpha \in (0,N)$, $\gamma,\mu,c \gt 0$ and $2_{\alpha} := \frac{N+\alpha}{N} \lt q \leq 2_{\alpha,s}^{\ast} := \frac{N+\alpha}{N-2s}$. $I_{\alpha}$ is the Riesz potential and $\lambda \in \mathbb{R}$ appears as an unknown Lagrange multiplier. By precisely restricting parameters $\gamma$, $\mu$ and $c$, using constrained variational method and introducing new relevant arguments, we establish several existence and nonexistence results. In particular, we consider the case $q = 2_{\alpha,s}^{\ast}$ which corresponds to equations involving double critical exponents, and the Hardy–Littlewood–Sobolev subcritical approximation method is used to solve the case.