Codice  QDD222 
Titolo  Normalized bound states for the nonlinear SchrÃ¶dinger equation in bounded domains 
Data  20160716 
Autore/i  Pierotti, D.; Verzini, G. 
Link  Download full text  Pubblicato  Calculus of Variations and PDEs 
Abstract  We investigate the standing wave solutions with prescribed mass (or charge) of the nonlinear SchrÃ¶dinger equation with power nonlinearity, in a bounded domain of dimension N and with Dirichlet boundary condition. Assuming that the exponent p>1 of the nonlinear term is Sobolevsubcritical, it follows by the GagliardoNirenberg inequality that there are solutions of any positive mass whenever p is less than the critical value 1+4/N. If p is equal or larger than 1+4/N, we prove that there are solutions having Morse index bounded above (by some positive integer k) only for sufficiently small masses. Lower bounds on these intervals of allowed masses are then obtained, by suitable variational principles, in terms of the Dirichlet eigenvalues of the Laplacian. 
