A Unified Approach to Singularly Perturbed Quasilinear Schrödinger Equations

In this paper we present a unified approach toinvestigate existence and concentration of positive solutions for the following class of quasilinear Schrödingerequations,-ε2Δu+V(x)u∓ε2+γuΔu2=h(u),x∈RN,where N⩾ 3 , ε> 0 , V(x) is a positive external potential,h is a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameter γ> 0. Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution uε,γ concentrating, as ε→ 0 , around minima points of the potential.