Standing waves with a critical frequency for a quasilinear Schrodinger equation

We consider the following quasilinear Schrodinger equation -epsilon(2)Delta omega + V(x)omega - epsilon(2)Delta(omega(2))omega = h(omega), omega > 0, x is an element of R-N. where N >= 4, lim V(x) > inf(vertical bar x vertical bar)(->infinity) V(x) > inf(x is an element of RN) V(x) = 0, and h satisfies a weaker growth condition than the Ambrosetti-Rabinowitz type condition in Byeon and Wang [Standing waves with a critical frequency for nonlinear Schrodinger equations, Arch. Ration. Mech. Anal. 165(4) (2002) 295-316; Standing waves with a critical frequency for nonlinear Schrodinger equations, II, Calc. Var. 18(2) (2003) 207-219]. We obtain the existence of the localized bound state solutions concentrating at an isolated component of the local minimum of V and whose amplitude goes to 0 as epsilon -> 0.