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

【摘要】 In this paper,we study the following generalized quasilinear Schr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∈R^N,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 smallλ > 0 using a change of variables and variational method.更多还原

【Abstract】 In this paper,we study the following generalized quasilinear Schr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∈R^N,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 smallλ > 0 using a change of variables and variational method.更多还原