Standing waves for quasilinear Schrِdinger equations

We study the existence of standing waves for a class of quasilinear Schrödinger equations − Δ u + V ( x ) u − [ Δ ( u 2 ) ] u = μ | u | q − 2 u + | u | r − 2 u , x ∈ R N , where N ≥ 3 , 2 < q < 4 , 4 < r < 2 ( 2 ∗ ) ≔ 4 N / ( N − 2 ) and μ ≥ 0 . V is a positive potential. One main difficulty in dealing with this problem seems to be that of obtaining the boundedness of a (PS) sequence for the corresponding functional. We overcome this difficulty by using Jeanjean’s result [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer type problem, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787–809]. Our results answer some questions raised in [J. M. do Ó, U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Comm. Pure Appl. Anal. 8 (2009) 621–644].