Multiplicity Of Solutions For Critical Quasilinear Schr(?)dinger Equations

The quasilinear Schr(?)dinger equation has a wide range of applications in plasma physics,fluid mechanics,glaciology,population dynamics and other disciplines,the study of this type of equation has important practical significance.Therefore,in recent decades,the research on the critical quasilinear Schr(?)dinger equation has been a hot spot in the field of nonlinear analysis and its applications in elliptic equations.This thesis mainly concerns the multiplicity of solutions for some critical quasilinear Schr(?)dinger equations in the whole space,including the modified Schr(?)dinger equation with bounded coefficient functions,the modified Schr(?)dinger equation with unbounded coefficient functions,and the critical-Laplacian equation.Firstly,we study the modified Schr(?)dinger equation with bounded coefficient functions.In order to obtain the existence of infinitely many solutions,we introduce the truncation equation,and prove that the truncation equation has infinitely many solutions by the method of regularization perturbation.Then,combined with the local Pohozeav identities,we obtain that any solutions corresponding to a sequence of truncation equations whose truncation parameter tends to zero satisfy the uniform decay estimate.From this,we can prove that the truncated equation and the original one share some common solutions,more and more as the truncation parameter tends to zero.Thereby,we prove that the modified Schr(?)dinger equation with bounded coefficient functions has infinitely many solutions.Furthermore,we consider the modified Schr(?)dinger equation with unbounded coefficient functions.In order to overcome the new difficulties caused by the unbounded coefficient functions,we introduce suitable variable substitution to transform the original equation into an equation with bounded coefficient functions.As a compensation,the low-order terms of the corresponding equation become very complicated.In order to deal with it,we carry out some precise estimates.And then combining the argument of concentrated compactness,we obtain multiple solutions of the modified Schr(?)dinger equation with unbounded coefficient functions.Finally,we discuss the p-Laplacian equation with critical exponents.Through the penalization method combined with truncation techniques and blow up analysis,we can prove that for any k,there exists e_k,such that for 0?e?e_k,the corresponding p-Laplacian equation admits at least k pairs of sign-changing solutions.Moreover,these solutions can only concentrate on a specific set of critical points of the potential function,in particular,on a local minimum point of the potential function.