Existence Of Solution And Multiple Solutions For A Class Of Quasilinear Schr(?)dinger Equation

In this paper, we consider the existence of nontrivial and multiple solutions for a class of following quasilinear Schrodinger equation where N≥3.g(t):Râ†'R+(R+=[0,+∞)) is a C1nondecreasing positive function with respect to|t|and the potential V:RNâ†'R is uniformly positive function.Firstly, using a change of variable, the existence of nontrivial solution for a class of quasilinear Schrodinger equation (Pi) is established based on the Mountain pass lemma and Lions lemma without (AR) condition.Next, we consider the existence of multiple solutions for a class of quasilinear Schrodinger equation with a non-homogenous perturbation where h(x)∈L2(RN). By Ekeland variational principle, we show that problem (P2) has a local minization solution; it follows from the Jeanjean result that problem (P2) has a mountain pass solution. Therefore, we have that problem (P2) possesses at least two positive solutions.