Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal

We consider quasilinear stationary Schrödinger equations of the form −∆u−∆(u2)u = g(x, u), x ∈ RN. (0.1) Introducing a change of unknown, we transform the search of solutions u(x) of (0.1) into the search of solutions v(x) of the semilinear equation −∆v = 1p 1 + 2f2(v) g(x, f(v)), x ∈ RN, (0.2) where f is suitably chosen. If v is a classical solution of (0.2) then u = f(v) is a classical solution of (0.1). Variational methods are then used to obtain various existence results. ∗Primary: 35J60, Secondary: 58E05.