@article{publications35021,
          volume = {342},
          number = {7},
          author = {Pascal B{\'e}gout and Jesus Ildefonso Diaz},
           title = {On a nonlinear Schr{\"o}dinger equation with a localizing effect},
       publisher = {Elsevier},
         journal = {Comptes rendus. Mathematique},
           pages = {459--463},
            year = {2006},
             url = {https://publications.ut-capitole.fr/id/eprint/35021/},
        abstract = {We consider the nonlinear Schr{\"o}dinger equation associated to a singular potential of the form 
\$a{\ensuremath{|}}u{\ensuremath{|}}{\^{ }}\{-(1-m)\}u+bu,\$ for some \$m{$\backslash$}in(0,1),\$ on a possible unbounded domain. We use some suitable energy methods to prove that if \${$\backslash$}mathrm\{Re\}(a)+{$\backslash$}mathrm\{Im\}(a){\ensuremath{>}}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{\ensuremath{>}}0.\$ This property contrasts with the behavior of solutions associated to regular potentials \$(m{$\backslash$}ge1).\$ Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential \$a{\ensuremath{|}}u{\ensuremath{|}}{\^{ }}\{-(1-m)\}u.\$ The existence of solutions is obtained by some compactness methods under additional conditions.}
}