Variational methods for nonlinear elliptic partial differential equations with nonlocal terms

Abstract

In this thesis, existence of standing waves for the Davey?Stewartson (DS) andgeneralized Davey?Stewartson (GDS) systems are established using variational methods.Since both the DS system and the GDS system reduce to a non-linear Schr¨odinger(NLS) equation with the only difference in their non-local term, arguments used in thisthesis apply to a larger class of equations which include the DS and GDS systems asspecial cases. Existence of standing waves for an NLS equation is investigated in twoways: by considering an unconstrained minimization problem and a constrained minimizationproblem. These two variational methods apply to the GDS system as welland here the sufficient conditions on the existence of standing wave solutions for theGDS system which are imposed by these methods and the minimizers obtained areinvestigated in comparison.