TitoloNormalized bound states for the nonlinear Schrödinger equation in bounded domains
Autore/iPierotti, D.; Verzini, G.
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PubblicatoCalculus of Variations and PDEs
AbstractWe investigate the standing wave solutions with prescribed mass (or charge) of the nonlinear Schrödinger 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 Sobolev-subcritical, it follows by the Gagliardo-Nirenberg 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.