Stability Of Standing Waves For Nonlinear Fractional Schr?dinger Equation With Coulomb Potential

The fractional nonlinear Schrodinger equation is a mathematical model,which can describe many physical phenomena.It is widely used in the fields of fluid mechanics,quantum mechanics,boson stars and water wave dynamics.The main purpose of this thesis is to investigate the stability of the standing waves for the nonlinear fractional Schrodinger equation with Coulomb potential in n-dimensional(n≥ 2)space.Firstly,we consider the well-posedness for the equation.Based on radial Strichartz estimation and contraction mapping principle,we prove the local well-posedness of Cauchy problem with radial initial data.Furthermore,the global existence of solutions in L2-subcritical and L2-critical cases are respectively obtained by combining Gagliardo-Nirenberg inequality and mass-energy conservation law.Secondly,we study the strong instability of the standing waves in the repulsive Coulomb potential.We use the compactness of the radial Sobolev embedding to prove the existence of the radial ground state.And then,a corresponding invariant evolution set to this equation is constructed by the convexity of mass-energy functional.Then,by applying with the local radial virial estimations,we obtain that the solutions of this equation blow up in a finite time.Furthermore,we show the strong instability of the standing waves in L2-supercritical case.Finally,we discuss the orbital stability of the standing waves for the repulsive Coulomb potential case.More specifically,we get the existence of the standing wave with non-radial initial data by using profile decomposition in Hs.Thus,the orbital stability of standing waves in L2-critical case is obtained via the corresponding minimization problem.