Quasilinear Schrodinger equations involving singular potentials

For the quasilinear Schrodinger equation -Delta u + V(x)u + k/2 Delta(u(2))u = h(u), u is an element of H-1 (R-N), where N >= 3, k is a real parameter, V(x) = V(vertical bar x vertical bar) 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 k < 0 and the existence of more and more radial solutions as k down arrow 0 . In the case k < 0, we allow h(u) = vertical bar u vertical bar(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 is an element of [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 k >= 0, our result extends and covers several related results in the literature, including the result of Berestycki and Lions.