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 − Δ u − κ u Δ ( u 2 ) + λ u = ( ∣ x ∣ − μ * F ( u ) ) f ( u ) , in R N , -\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 ≥ 3 N\ge 3 , μ ∈ ( 0 , N ) \mu \in \left(0,N) , F ( t ) = ∫ 0 t f ( s ) d s F\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 L 2 {L}_{2} -norm of the solution is not confirmed and the L 2 {L}_{2} -norm of the solution is prescribed. Under the almost optimal assumptions on f f , we obtain the existence of a sequence of radial solutions for two cases.