Existence And Asymptotic Properties Of Solutions For Qusi-linear Schr(o|¨)dinger Equation

Quasi-linear Schr(o|¨)dinger equation is derived from physical models such as plasma physics and dissipative quantum mechanics.In recent years,much attention was paid to the existence and asymptotic properties of solutions for quasi-linear Schr(o|¨)dinger equation.This paper mainly studies a class of quasi-linear Schr(o|¨)dinger equations which comes from high-power ultra-short laser materials.From the variational point of view,the energy functional corresponding to this type quasi-linear Schr(o|¨)dinger equation is not well-defined in the usual Sobolev space H1(RN).This causes some mathematical difficulties which make the study of quasi-linear Schr(o|¨)dinger equation particularly interesting.By using the change of variables and critical point theory,the existence results of non-trivial solutions when the nonlinear terms satisfying(AR)conditions have been deduced for quasi-linear Schr(o|¨)dinger equation.In this paper,the existence and asymptotic behavior of nontrivial solutions are studied when the non-linear term does not satisfy the(AR)condition.The specific researches and innovations are as follows:Firstly,the existence of ground state solution is considered in constant potential.This result extend the result of classical Berestyck-Lions'field equation.Furthermore,we know that the(AR)condition is a necessary condition for the proof of the existence.In this chapter,the existence of nontrivial solutions for quasi-linear Schr(o|¨)dinger equation with bounded potential is obtained by using the change of variables and critical point theory when the non-linear term does not satisfy the(AR)condition.Secondly,we study the existence and asymptotic behavior of positive radial symmetric solutions for a general quasi-linear Schr(o|¨)dinger equation.By discussing some properties of the change variables under different parameters,the existence,radial symmetry and exponential decay of the ground state solution are obtained by using the idea of constraint variation.Furthermore,for this general quasi-linear Schr(o|¨)dinger equation,the asymptotic behavior of the ground state is much less studied.In this chapter,using the Morse iteration method,the uniformly L? estimated of the ground state for quasi-linear Schr(o|¨)dinger equation is obtained.Then,the asymptotic behavior of ground state solution when the coefficient of quasi-linear term tends to zero is get.Finally,we give some summarizes and expectations.Especially,there is no result respect to L2-constrained solution for this type quasi-linear Schr(o|¨)dinger equation.Therefore,some Gagliardo-Nirenberg inequalities are obtained since they are necessary for the existence of L2-constrained solution.Using these inequalities,we hope that we can give a complete classification with respect to the exponent in the nonlinear term for its L2-constraint solutions.