Quasilinear Schrödinger equations involving singular potentials

Abstract For the quasilinear Schrödinger equation − Δ u + V ( x ) u + κ 2 Δ ( u 2 ) u = h ( u ) , u ∈ H 1 ( R N ) , where N ⩾ 3, κ is a real parameter, V(x) = V(|x|) is a potential allowed to be singular at the origin and h : R → R is a nonlinearity satisfying conditions similar to those in the paper (1983 Arch. Ration. Mech. Anal. 82 347–375) by Berestycki and Lions, we establish the existence of infinitely many radial solutions for κ < 0 and the existence of more and more radial solutions as κ ↓ 0. In the case κ < 0, we allow h(u) = |u| p−2 u for p in the whole range (2, 4N/(N − 2)) and this is in sharp contrast to most of the existing results which are only for p ∈ [4, 4N/(N − 2)). Moreover, our result in this case extends the result of Berestycki and Lions in the paper mentioned above to quasilinear equations with singular potentials. In the case κ ⩾ 0, our result extends and covers several related results in the literature, including the result of Berestycki and Lions.