On The Solutions Of Two Classes Of Quasilinear Schrodinger Equations

With the infiltration of mathematics in various disciplines,some new forms of Schrodinger equations are proposed to describe the changing processes of physics,finance and biology.Therefore,It is a great significance to study the solutions of various Schrodinger equations and has always been a hot topic.The main purpose of this dissertation is to discuss the existence,multiplicity and concentration of solutions for two classes of nonlinear Schrodinger equations by variational method.The dissertation is divided into three parts,the main contents are as followsIn the first part,we first introduce the backgrounds and research status of the problems combining the two classes of Schrodinger equation to be studied,then give the main results.In the second part,the existence of ground state solutions for the following fractional Schrodinger equation(-?)?u+?u+[(-?)?u2]u=?|u|p-1u+|u|q-2u,X?RN,is concerned,where ?>0,N?3,?>0,??(0,1),2<p+1<q?2N/N-2?:=2?*and(-?)? denotes the fractional Laplacian of order ?.Our research shows that:As 2<p+1<q<2?*,for any ?>0,the above equation possesses a ground state positive solution.As q=2?*,there exists a ?*>0 such that as ???*,the above equation possesses a ground state solution,tooOur results are a meaningful glance to this aspect of researchIn the third part,the following quasilinear Schrodinger equations-?2?u+V(x)u-?2?A(u2)u=|u|22*-2u+g(u),x ?RN.is discussed,and the existence,multiplicity and concentration of positive solutions are given by add global and local conditions to the potential function V,respectively.Where N?3,? is positive parameters and 2*=2N/N-2 is the critical exponent,V?C(RN,R+),g?C(R,R).Our research shows that:under some suitable assumptions on the g,no matter the global condition or the local condition on the potential V is satisfied,the above equation has at least cat??(?)positive solutions.Moreover,if u? denotes one of these solutions and ???RN is its global maximum,then lim??0 V(?)=V0.Where?:={x ?RN:V(x)=V0} and ??:={x?RN:d(x,?)??}.These results weaken the conditions imposed on the function g,and these results are applicable to more objects.Furthermore,these results are new even in the semilinear case.