| The self-channel problem of high-power ultrashort laser pulses in plasma is often described by the relativistic nonlinear Schr?dinger equation.Since the energy functional corresponding to the equation is not smooth,the critical point theory can’t be directly applied.The existing research results on this equation are mainly aimed at the existence of the solutions when the power function is nonlinear.In this thesis,we will discuss the existence and non-existence of radial ground state solutions(that is,the solution with minimum energy among all non-trivial radial solutions)for a class of relativistic nonlinear Schr?dinger equations with general nonlinear terms by using the mountain road theorem,Strauss compactness lemma and Pohozaev identity.For the existence of the radial ground state solutions of the equation,this thesis mainly considers the case where the general nonlinear terms satisfy the classical Berestycki-Lions condition.In this thesis,the variable substitution technique is used to convert the relativistic nonlinear Schr?dinger equation into a semi-linear elliptic equation,so as to overcome the difficulty of the non-smooth energy functional of the equation.However,because the equation contains the potential terms,the transformed nonlinear terms no longer satisfy the Berestycki-Lions condition.Therefore,this thesis combines the mountain road theorem,Strauss compactness lemma and perturbation techniques to prove the existence of the radial ground state solutions of the semi-linear elliptic equation,and further combines with the regularity estimation to obtain the existence of the radial ground state solutions of the relativistic nonlinear Schr?dinger equation.For the non-existence of the solutions of the relativistic nonlinear Schr?dinger equation containing general nonlinear terms,this thesis first establishes the corresponding Pohozaev identity of the equation,and then,by using constrained variational techniques and proof by contradiction,the non-existence of non-trivial solutions of the equation is obtained when the potential function satisfies the given conditions.To sum up,in this thesis,the existence and non-existence of the radial ground state solutions of the relativistic nonlinear Schr?dinger equation with radial vanishing potential are obtained when the nonlinear terms satisfy the Berestycki-Lions condition.By introducing the Strauss compactness lemma and perturbation techniques,the results of the semi-linear Schr?dinger equation are extended to the relativistic nonlinear Schr?dinger equation,thus enriching the results of the relativistic nonlinear Schr?dinger equation and extending the application of the critical point theory to nonsmooth functionals. |
