On a star graph , we consider a nonlinear Schrödinger equation with focusing nonlinearity of power type and an attractive Dirac's delta potential located at the vertex. The equation can be formally written as , where the strength α of the vertex interaction is negative and the wave function Ψ is supposed to be continuous at the vertex. The values of the mass and energy functionals are conserved by the flow. We show that for the energy at fixed mass is bounded from below and that for every mass m below a critical mass it attains its minimum value at a certain .Moreover, the set of minimizers has the structure . Correspondingly, for every there exists a unique such that the standing wave is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is nontrivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to .
@article{AIHPC_2014__31_6_1289_0, author = {Adami, Riccardo and Cacciapuoti, Claudio and Finco, Domenico and Noja, Diego}, title = {Constrained energy minimization and orbital stability for the {NLS} equation on a star graph}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1289--1310}, publisher = {Elsevier}, volume = {31}, number = {6}, year = {2014}, doi = {10.1016/j.anihpc.2013.09.003}, mrnumber = {3280068}, zbl = {1304.81087}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.09.003/} }
TY - JOUR AU - Adami, Riccardo AU - Cacciapuoti, Claudio AU - Finco, Domenico AU - Noja, Diego TI - Constrained energy minimization and orbital stability for the NLS equation on a star graph JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 1289 EP - 1310 VL - 31 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.09.003/ DO - 10.1016/j.anihpc.2013.09.003 LA - en ID - AIHPC_2014__31_6_1289_0 ER -
%0 Journal Article %A Adami, Riccardo %A Cacciapuoti, Claudio %A Finco, Domenico %A Noja, Diego %T Constrained energy minimization and orbital stability for the NLS equation on a star graph %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 1289-1310 %V 31 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.09.003/ %R 10.1016/j.anihpc.2013.09.003 %G en %F AIHPC_2014__31_6_1289_0
Adami, Riccardo; Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego. Constrained energy minimization and orbital stability for the NLS equation on a star graph. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 6, pp. 1289-1310. doi : 10.1016/j.anihpc.2013.09.003. http://www.numdam.org/articles/10.1016/j.anihpc.2013.09.003/
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