The Existence Of Solution For Generalized Quasilinear Schr(?)dinger And Quasilinear Schr(?)dinger Equation

Studying the existence of solutions to the Schrodinger equation is important for solving the wave functions in quantum mechanics and the distribution of statistical particles in space.In this paper,we mainly apply minimax methods in nonlinear functional analysis to study the existence and concentration behavior of solutions to the generalized quasilinear and quasilinear Schrodinger equations.The main contents are as follows:The first chapter introduces the research background,research significance and research status of the problem.The main research work and relevant preparatory knowledge of this paper are elaborated.The second chapter studies the generalized quasilinear Schrodinger equation with the Choquard nonlinear term:-div（g²（u）▽u）+g（u）g’（u）|▽u| ²+V（x）u=（I_α*|u|^p）|u|^p-2u,x∈R^N,where N≥3,0<α<N,2（N+α）/N<p<2（N+α）/（N-2）,V（x）is a potential function,I_α is a Riesz potential.When the potential function is a periodic function of the variable（i=1,…,N）,we prove the existence of the positive solution of the equation.The third chapter studies the quasilinear Schrodinger equation with steep well potential:where N≥3,λ>0,12-（?）<p<2*,V（x）is a potential function,Ω:=V^-1（0）has nonempty interior.We prove the existence and concentration behavior of nontrivial solutions of the equation using the mountain pass theorem.The fourth chapter studies the quasilinear Schrodinger equation with critical exponents:where N≥3,V（x）is a potential function,f is a nonlinear term.The existence of ground state solutions of this equation is demonstrated with the help of Pohozaev manifold and constrained minimization.The fifth chapter presents the conclusions and outlook in this paper.