Semiclassical ground states for quasilinear Schrödinger equations with three times growth

In this paper, we study the quasilinear Schrödinger equation−ϵ2Δu+V(x)u−ϵ2uΔ(u2)=Q(x)u3,u∈H1(R3), where ϵ>0 is a parameter, V and Q are positive bounded functions. For the equation with three times growth, we establish the existence of ground states for ϵ small using the method of Nehari manifold. We also describe the concentration phenomena of ground states as ϵ→0.