EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHRDINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS

In this paper,we study the following generalized quasilinear Schrdinger equations with critical or supercritical growths-div(g~2(u)▽u) + g(u)g′(u)|▽u|~2+ V(x)u = f(x,u) + λ|u|p-2u,x∈R~N,where λ>0,N≥3,g:R → R~+ is a C~1 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 smallλ > 0 using a change of variables and variational method.