Existence of Nontrivial Solutions for Generalized Quasilinear Schr?dinger Equations with Critical Growth

摘要

We study the following generalized quasilinear Schr?dinger equations with critical growth -div(g~2(u)?u) + g(u)g'(u)|?u|~2 + V(x)u = λf(x, u) + g(u)|G(u)|~(2*-2)G(u), x ∈ R~N, where λ > 0, N ≥ 3, g(s) : R → R~+ is a C~1 even function, g(0) = 1, and g'(s) ≥ 0 for all s ≥ 0, where G(u):=∫_0~u g(t)dt. Under some suitable conditions, we prove that the equation has a nontrivial solution by variational method.