Research On Existence Of Standing Wave Solutions For Generalized Quasilinear Schr(?)dinger Equations

With the continuous development of science and technology,the application of the quasilinear Schr(?)dinger equation has become more and more extensive,and therefore it has become an important research object for many scholars.In this thesis,two kinds of generalized quasi-linear Schr(?)dinger equations are studied by variational method.The basic idea is to transform boundary value problems of differential equations into variational problems on variational method,so as to consider the existence and related properties of standing wave solutions of equations by analytical methods.The mainly study are as follows:In chapter one,we introduce the research background and current situation at home and abroad of the quasilinear Schr(?)dinger equation,along with expounding the main work of this thesis and the relevant basic knowledge.In chapter two,we discuss the existence and asymptotical behavior of ground state solutions to the generalized quasilinear Schr(?)dinger equation (?) which is equipped with deep well potential and critical growth exponent.In this equation,we assume that α≥1,N≥3,g:R→R+ is an even differentiable function,h:RN×R→R is a continuous function,and the potential a(x):RN→R is continuous in RN.In chapter three,we study the existence of radial solutions and many non-radial solutions to the generalized quasilinear Schr(?)dinger equation-div(g2(u)▽u)+g(u)g’(u)|▽u|2+V(|x|)u=f(|x|,u),x∈RN,Where N≥2,g:R→R+ is an even differentiable function and f:[0,+∞)×R→R is a continuous function.In chapter four,we summarize the main research methods and conclusions of this thesis,and put forward the direction of future research needs to be improved and continued.