Orbital Stability Of Standing Waves For The Nonlinear Schr(?)dinger Equation With Mixed Nonlinearities

The aim of this thesis is to study the orbital stability of standing waves for the following nonlinear Schrodinger type equation with mixed power-type and Choquard-type nonlinearities (?) where N≥ 3,0<μ<N,a≥ 0,α≥ 0,2α+μ≤N,0<q<4/N,2-(2α+μ)/N<p<(2N-2α-μ)/(N-2)and ψ(t,x):[0,T)×R~N→C is the complex function with 0<T≤∞.Firstly,we derive the best constant for a generalized Gagliardo-Nirenberg inequality by considering a variational problem.Besides,in the mass-subcritical and mass-critical cases,i.e.,a>0,2-(2α+μ)/N<p≤(2+2N-2α-μ)/N,applying the idea of Cazenave and Lions,we prove that the set of energy minimizers is orbitally stable by considering the constrained global minimization problem.Finally,in the mass-supercritical case,i.e.,a>0,(2+2N-2α-μ)/N<p<(2N-2α-μ)/(N-2),applying the idea of Jeanjean,we prove the orbital stability of the set of energy minimizers by considering the constrained local minimization problem.