Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations

Abstract

This article is concerned with the existence of multiple normalized solutions for a class of Choquard equations with a parametric perturbation −ε2Δu+V(x)u=λu+ε−α(Iα*F(u))f(u),x∈RN,∫RN∣u∣2dx=a2εN,\left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V\left(x)u=\lambda u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }* F\left(u))f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2}{\varepsilon }^{N},\hspace{1.0em}& \end{array}\right. where a>0a\gt 0 is a constant, ε>0\varepsilon \gt 0 is a parameter, N≥3N\ge 3, α∈(0,N)\alpha \in \left(0,N), λ∈R\lambda \in {\mathbb{R}} is unknown and appears as a Lagrange multiplier, ff is a continuous function with L2{L}^{2}-subcritical growth, and V:RN→[0,∞)V:{{\mathbb{R}}}^{N}\to \left[0,\infty ) is a continuous function, satisfying del Pino and Felmer’s local conditions. With the help of the penalization method, and Lusternik-Schnirelmann theory, we investigate the relationship between the number of positive normalized solutions and the topology of the set, where the potential VV attains its minimum value if the parameter ε>0\varepsilon \gt 0 is small.

Keywords

• choquard equations
• normalized solutions
• lusternik-schnirelmann category
• variational methods
• 35a15
• 35b33
• 35j20
• 35j60