Positive Solutions,Negative Solutions And Sign-Changing Solutions For Several Quasilinear Schr(?)dinger Equations

With the in-depth study of science and technology,Schr???dinger equations has become an important research object for mathematicians and physicists,and has far-reaching influence on nonlinear analysis,differential geometry and mathematical physics,etc.The study of quasilinear problems has received much attention in recent years.In this paper,some tools and analytical techniques in the variational and critical point theory are used to study the existence of solutions for two quasilinear Schr???dinger equations.Details as follows:The first one,we study this form of quasilinear Schr???dinger equation:-?u+V?x?u-₂¹??|u|²?u=g?x,u?,x?R^N,and where V?C?R^N,R?,g?C?R^N×R,R?.Using the mountain lemma under some specific conditions,we can get a positive solution and a negative solution to the problem.In addition,constraint minimization theory,we can get another different positive solution and negative solution to the problem.The second one,we study the existence of sign-changing solutions for the fol-lowing quasilinear Schr???dinger equation:-(1+??R³(1+2?|u|^2?2?-1?)|?u|²)[?u+?(|u|^2?)|u|^2?-2u]+V?x?u=g?x,u?,x?R³,in the whole three-dimension,where 1??>₂¹,V?x?is a potential function,K₁?x?h?u??g?x,u??K₂?x?h?u?.The non-Nehari manifold method and deforma-tion lemma are used to obtain the existence of the solution of the problem.There are three chapters in this paper.The first chapter is the introduction,which mainly introduces the research background and preliminary knowledge of the quasilinear Schr???dinger equation.The second chapter studies the existence and mul-tiple solutions of the first quasilinear Schr???dinger equation by using the mountain lemma method.The third chapter uses the non-Nehair method to study the exis-tence of the sign-changing solution of the second quasilinear Schr???dinger equation.