%A Pascal Bégout
%A Jesus Ildefonso Diaz
%J Comptes rendus. Mathematique
%T On a nonlinear Schrödinger equation with a localizing effect
%X We consider the nonlinear Schrödinger equation associated to a singular potential of the form 
$a|u|^{-(1-m)}u+bu,$ for some $m\in(0,1),$ on a possible unbounded domain. We use some suitable energy methods to prove that if $\mathrm{Re}(a)+\mathrm{Im}(a)>0$ and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any $t>0.$ This property contrasts with the behavior of solutions associated to regular potentials $(m\ge1).$ Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential $a|u|^{-(1-m)}u.$ The existence of solutions is obtained by some compactness methods under additional conditions.
%N 7
%P 459-463
%V 342
%D 2006
%I Elsevier
%L publications35021