EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHR(O)DINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS

摘要

In this paper,we study the following generalized quasilinear Schr(o)dinger 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 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.