Existence Of Solutions To A Class Of Quasilinear Critical Schrodinger Equations With A Parameter

The quasilinear Schr(?)dinger equation is derived from plasma physics and is a problem that has received widespread attention in the field of nonlinear analysis in recent years.This thesis mainly uses the variational method to study the existence of solutions for a class of quasilinear Schr(?)dinger equations with Sobolev critical exponents.Firstly,we introduce the background and recent research progress of the quasilinear Schr(?)dinger equation.Secondly,we study the existence of solutions to the following critical quasilinear Schr(?)dinger equations in 2-dimensions-?u+V(x)u-???(|u|2?)|u|2?-2=f(u),x?R2,where the parameters ??(1/2,1),??R are constants,V(x)is a given potential function,and f has critical growth at infinity.Using variable substitution,combining the Trudinger-Moser inequality and the mountain pass theorem,we prove that the equation has at least one non-trivial solution under appropriate conditions.Finally,in high-dimensional cases(dimensions N?3),we study the existence of solutions to the following semi-classical problems of critical quasilinear Schr(?)dinger equations-?2?u+V(x)u-???2?(|u|2?)|u|2?-2=|u|q-2u+|u|2*-2u,x?RN where 0<?<<1,??R are constants,q?(2,2*),2*=2N/N-2 is the Sobolev critical exponent.In the case of ??(0,1/2),we use the variable substitution and the truncation methods to obtain the existence of positive solutions.