Multiple solutions for quasi-linear elliptic equations with Berestycki-Lions type nonlinearity

We studied the modified nonlinear Schro<spacing diaeresis>dinger equation - Delta u - 1 2 Delta( u 2 ) u = g ( u ) + h ( x ) , u is an element of H 1 ( R N ) , (0.1) where N >= 3, g is an element of C ( R , R) is a nonlinear function of Berestycki-Lions type, and h t 0 is a nonnegative function. When II h II L 2 ( R N ) is suitably small, we proved that (0.1) possesses at least two positive solutions by variational approach, one of which is a ground state while the other is of mountain pass type.