Existence of Nontrivial Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

We study the following generalized quasilinear Schrödinger equations with critical growth -divg2u∇u+gug′u∇u|2+Vxu=λfx,u+guGu|2⁎-2Gu,x∈RN, where λ>0, N≥3, g(s):R→R+ is a C1 even function, g(0)=1, and g′(s)≥0 for all s≥0, where G(u)≔∫0ug(t)dt. Under some suitable conditions, we prove that the equation has a nontrivial solution by variational method