The Existence Of Positive Solutions For Two Classes Of Quasilinear Schr(?)dinger Equations

In recent years,the existence of solution for quasilinear Schr(?)dinger equations has been attended by many researchers.Quasilinear Schr(?)dinger equations relate to the fields of quantum mechanics,theoretical physics,fluid mechanics,etc.It plays an important role in variational methods,topology,differential geometry and some other mathematical disciplines.In this thesis,we mainly discuss the existence of positive solutions for two kinds of quasilinear Schr(?)dinger equations.This thesis consists of four chapters.In the first chapter,we give some preliminary knowledge,including some symbolic descriptions,definitions,lemmas,theorems and several inequalities.In the second chapter,we study the existence of positive solutions of the following quasilinear Schr(?)dinger equations(Ⅰ)in RN(N≥2).-Δu+u-2[Δ(|u|2)]u=θ|u|p-2u+λk(x)u,x∈RNN.(Ⅰ)By changing the variables,we transform the functional J(u)which may not be defined in the space H1(RN)to the new functional I(v)which is well defined in H1(RN).Therefore,we obtain a positive solution of the equation under the suitable conditions by the mountain pass lemma and the strong maximum principle.In the third chapter,we continuously examine existence of positive solutions of the following quasilinear Schr(?)dinger equations(Ⅱ).-Δu+u-[Δ|u|2α]|u|2α-2=θ|u|p-2u+λk(x)u,x∈RN.(Ⅱ)On the basis of the previous chapter,we similarly obtain the existence of positive solutions for a class of quasilinear Schr(?)dinger equations in a more general casel(s)=sα,α>1 in RN(N≥2).The fourth chapter includes summary and outlook.We summarize the main contents of the thesis and point out the flow-up research directions for further research.