Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities

Abstract

In this article, we study the following quasilinear equation with nonlocal nonlinearity −Δu−κuΔ(u2)+λu=(∣x∣−μ*F(u))f(u),inRN,-\Delta u-\kappa u\Delta \left({u}^{2})+\lambda u=\left({| x| }^{-\mu }* F\left(u))f\left(u),\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where κ\kappa is a parameter, N≥3N\ge 3, μ∈(0,N)\mu \in \left(0,N), F(t)=∫0tf(s)dsF\left(t)={\int }_{0}^{t}f\left(s){\rm{d}}s, and λ\lambda is a positive constant. We are going to analyze two cases: the L2{L}_{2}-norm of the solution is not confirmed and the L2{L}_{2}-norm of the solution is prescribed. Under the almost optimal assumptions on ff, we obtain the existence of a sequence of radial solutions for two cases.

Keywords

• quasilinear choquard equation
• multiple solutions
• berestycki-lions nonlinearities
• 35q40
• 35j50
• 58e05