Multiplicity of normalized solutions for nonlinear Choquard equations

Abstract

In this paper, we consider the following nonlinear Choquard equation with prescribed L 2-norm: −Δu+λu=Iα∗F(u)f(u) in RN,∫RN|u|2dx=a>0,u∈H1(RN), $\begin{cases}-{\Delta}u+\lambda u=\left({I}_{\alpha }\ast F\left(u\right)\right)f\left(u\right) \,\text{in}\, {\mathbb{R}}^{N},\quad \hfill \\ {\int }_{{\mathbb{R}}^{N}}\vert u{\vert }^{2}\mathrm{d}x=a{ >}0, u\in {H}^{1}\left({\mathbb{R}}^{N}\right),\quad \hfill \end{cases}$ where N≥3,α∈(0,N),Iα(x)=Aα|x|N−α $N\ge 3,\alpha \in \left(0,N\right),{I}_{\alpha }\left(x\right)=\frac{{A}_{\alpha }}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, f∈C(R,R) $f\in C\left(\mathbb{R},\mathbb{R}\right)$ , F(s)=∫0sf(t)dt $F\left(s\right)={\int }_{0}^{s}f\left(t\right)\mathrm{d}t$ and λ is an unknown Lagrange multiplier. Under the general assumption of F and within an appropriate mass range, we prove the existence and multiplicity of solutions to this problem, which may manifest as global minimizer, local minimizer, or mountain pass-type solutions.

Keywords

• choquard equation
• normalized solution
• concave-convex nonlinearity
• variational methods
• 35j20
• 35b33
• 35b45