Quasilinear Schrödinger equations with bounded coefficients

Abstract We study quasilinear elliptic equations of the form − div A ( u ) ∇ u + 1 2 A ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( u ) , u ∈ H 1 ( R N ) , where N ⩾ 3 , A ∈ C 1 ( R , R + ) is a bounded function, V(x) is allowed to be singular at the origin, and h ∈ C ( R , R ) is a general nonlinearity. Such type of equations has been derived as models of several physical phenomena corresponding to various types of A . Despite the lack of a priori compactness condition for the energy functional, we develop a new variational approach to deal with the issues of multiple radial solutions, nonradial solutions and sign-changing solutions.