Existence and multiplicity of normalized solutions for a class of fractional Choquard equations

Abstract

In this paper, we study the existence and multiplicity of solutions with a prescribed ${L^2}$-norm for a class of nonlinear fractional Choquard equations in ${\R^N}$: where $N\geq3$, $s\in~(0,1)$, $\alpha~\in(0,N)$, $p\in(\max~\{~1~+~\frac{{\alpha~+~2s}}{N},2\},\frac{N+\alpha}{N-2s})$ and ${\kappa~_\alpha~}(x)~=~{~|~x~|^{\alpha~-~N}}$.To get such solutions, we look for critical points of the energy functional on the constraints 0. \end{equation}]]> For the value $p\in(\max~\{~1~+~\frac{{\alpha~+~2s}}{N},2\},\frac{N+\alpha}{N-2s})$ considered, the functional $I$ is unbounded from below on $S(c)$. By using the constrained minimization method on a suitable submanifold of $S(c)$, we prove that for any $c>0$, $I$ has a critical point on $S(c)$ with the least energy among all critical points of $I$ restricted on $S(c)$. After that, we describe a limiting behavior of the constrained critical point as $c$ vanishes and tends to infinity. Moreover, by using a minimax procedure, we prove that for any $c>0$, there are infinitely many radial critical points of $I$ restricted on $S(c)$.