Normalized Solutions for a Class of Fractional Choquard Equation With Potential and Combined Nonlinearities

Abstract

In this paper, we consider the following fractional Choquard equation: ${(-\Delta )}^{s}u+V\left(x\right)u+\lambda u=(I_{\alpha }\ast |u{|}^{2_{\alpha ,s}^{\ast }}\left)\right|u{|}^{2_{\alpha ,s}^{\ast }-2}u+\mu |u{|}^{q-2}u,x\in {ℝ}^N$, with prescribed mass ${\int }_{{ℝ}^N}|u{|}^{2}dx=a^2$, where $s\in (0,1),\mu >0,\frac{N}{2}>s,0<\alpha <\min \{N,4s\},I_{\alpha }$ is the Riesz potential, $V$ is an external potential vanishing at infinity and the parameter $\lambda \in ℝ$ arises as Lagrange multiplier. The purpose of this paper is to establish the existence of solutions with prescribed norm to this class of nonlinear equations. Under some $L^2$ -subcritical, $L^2$ -critical and $L^2$ -supercritical perturbation $\mu |u{|}^{q-2}u$, respectively, we obtain several existence results. By limiting the range of $\mu$, for $q\in (\bar{q},2_s^{\ast }]$, we prove that there exists a positive ground state normalized solution for the above problem with $a>0$. Furthermore, for $q\in (2,\bar{q})$, we prove that there exists $a_0>0$ such that the normalized solution with negative energy to the above problem can be obtained when $a\in (0,a_0)$.