Nonlinear Schrödinger equations with single power nonlinearity and harmonic potential

We consider a generalized nonlinear Schrodinger equation (GNLS) with a single power nonlinearity of the form lambda vertical bar phi vertical bar(p), with p > 0 and lambda is an element of R, in the presence of a harmonic confinement. We report the conditions that p and lambda must fulfill for the existence and uniqueness of ground states of the GNLS. We discuss the Cauchy problem and summarize which conditions are required for the nonlinear term lambda vertical bar phi vertical bar(p) to render the ground state solutions orbitally stable. Based on a new variational method we provide exact formulae for the minimum energy for each index p and the changing range of values of the nonlinear parameter lambda. Also, we report an approximate close analytical expression for the ground state energy, performing a comparative analysis of the present variational calculations with those obtained by a generalized Thomas-Fermi approach, and soliton solutions for the respective ranges of p and lambda where these solutions can be implemented to describe the minimum energy.