Existence of nontrivial solutions for generalized quasilinear Schrödinger equations with critical or supercritical growths

In this paper, we study the following generalized quasilinear Schrödinger equations with critical or supercritical growths -div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=f(x,u)+λ|u|p-2u,x∈RN,where λ>0,N≥3,g:R→R+ is a C1 even function, g(0)=1,g′(s)≥0 for all s≥0,lim|s|→+∞g(s)|s|α-1:β>0 for some α ≥ 1 and (α-1)g(s)>g′(s)s for all s > 0 and p ≥ α2*. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 using a change of variables and variational method.