On The Solutions Of Two Classes Of Quasi-linear Schr(?)dinger Equations

In this paper,we mainly study the following two classes of quasilinear Schr(?)dinger equations.The first one is to study the existence and multiplicity of normalized solutions of the following autonomous quasi-linear Schr(?)dinger equation (?)The other one is to study the existence of multiple solutions for the following nonautonomous quasi-linear p-Laplacian equation in the whole space RN-div(A(x,u)|▽u|p-2▽ u)+1/p At(x,u)|▽u|p+V(x)|u|p-2u=g(x,u).(0.0.4)In the process of studying,we will use the following three methods:the dual approach,the constrained minimization method and the variational perturbation method.When dealing with the problem of normalized solution of the autonomous quasilinear equation(0.0.3),we mainly discuss it in two cases:mass sub-critical(i.e.,p ∈(2.4+4/N))case,and mass critical and super-critical(i.e.,p ≥ 4+4/N)cases.For mass sub-critical case,by using the dual approach(see[46,92]),we first prove the multiplicity of normalized solutions of equation(0.0.3).There are three reasons for using the dual method:the first one is to overcome the inevitable difficulties when dealing with the quasi-linear equations.That is,the variational functional corresponding to the quasi-linear equations is non-differentiable.The second reason is that the constrained minimization method cannot be used to deal with the multiplicity results here.The third reason is that the perturbation method will no longer be applicable when studying the multiplicity of solutions in this case.Although the variational perturbation method can also be used to deal with the non-differentiable problem of quasi-linear functional,the energy levels corresponding to the solutions constructed by the perturbation method are approaching zero from below,which makes it difficult to distinguish the critical points corresponding to the "zero" energy level.For mass critical and super-critical cases,we found a new scaling transformation,which can be used to show that for p ∈[4+4/N,2N/N-2]with N=3,and for p ∈[4+4/N,∞)with N=1,2,the minimum of the functional corresponding to equation(0.0.3)is strictly decreasing with respect to A.From this we can prove that the minimum is achieved.Then,inspired by[93],we extend the existence range of ground state solution of the quasilinear equation(0.0.3)to ∈[4+4/N,2N/N-2]if N=3 and p E[4+4/N,∞)if N=1,2.In addition,for p E(2+4/N,4+4/N],through careful analysis,we can draw a conclusion that if the L2 constraint value A is in a precise constraint range,there does not exist any critical points of the functional corresponding to the equation(0.0.3).In the latter part,we will use the variational perturbation method to analyze the existence of unconstrained multiple solutions of the non-autonomous pLaplacian equation(0.0.4).In this process,in order to overcome the difficulty of non-differentiability of the functional,and ensure the compactness of the PS sequence,we will use the perturbation variational method to prove the existence of multiple solutions of the equation(0.0.3).It should be noted that there are not many studies on the existence of multiple solutions to the equation(0.0.4)at present.