acase@tulane.edu

acase@tulane.edu

The objective of this dissertation is to give a description of the non-degenerate invariant bilinear forms on finite-dimensional Lie algebras over fields of characteristic zero which do not generally arise as trace forms Chapter I provides the necessary background information on Lie algebras Chapter II introduces the notion of invariant bilinear forms, with special attention given to trace forms Chapter III is devoted to a general presentation of L-modules with a non-degenerate invariant bilinear form, particularly in relation to duality in the presence of a non-degenerate bilinear form. The first part of the chapter exploits the annihilator ideal in greater detail Chapter IV treats invariant bilinear forms on simple and semi-simple Lie algebras. For simple Lie algebras L, all non-zero invariant bilinear forms are non-degenerate and, if F is an algebraically closed field, every such form is an F-multiple of the Killing form (kappa) of L. It is shown that this result still holds if F is replaced by the field of real numbers Chapter V provides background information on nilpotent and solvable Lie algebras. Then it is shown that a solvable Lie algebra can carry a non-degenerate invariant bilinear form only if it is of a special type. The theory of ground ring extensions developed here will lead to a construction of nilpotent Lie algebras of arbitrary nilpotency class with a non-degenerate invariant bilinear form Chapter VI analyzes the structure of finite-dimension Lie algebras L over fields of characteristic zero with a non-degenerate invariant bilinear form. The results of this investigation lead to a description of the non-degenerate invariant bilinear forms on L / acase@tulane.edu

acase@tulane.edu

What are the roles of two competing potential functions $V(x)$ and $K(x)$ in the process of concentration for ground state solutions of an elliptic equation $h\sp2\Delta u-V(x)u+ K(x)\vert u\vert\sp{p-1}u=0,x\in R\sp{n}$ arising in the study of standing wave solutions to nonlinear Schrodinger equations? This is the motivating question and one of the questions this dissertation answers. After a careful analysis of movement of the energy, the existence and concentration behaviors of ground states are established and an explicit formula for the concentration points are found. Then the variational methods are adapted to attack an equation with more general nonlinear terms. Similar results are obtained and two limiting situations are discussed On another direction, it is proved that positive bound states for nonlinear Schrodinger equations exist and these solutions can form sequences concentrating at each critical point of a so-called minimax value function under certain technical conditions. At the end, a necessary condition for concentration of positive bound states is given / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu