On the existence of bound and ground states for some coupled nonlinear Schrödinger-Korteweg-de Vries equations Articles uri icon

publication date

  • November 2017

start page

  • 407

end page

  • 426

issue

  • 4

volume

  • 6

International Standard Serial Number (ISSN)

  • 2191-9496

Electronic International Standard Serial Number (EISSN)

  • 2191-950X

abstract

  • We show the existence of positive bound and ground states for a system of coupled nonlinear Schrödinger-Korteweg-de Vries equations. More precisely, we prove that there exists a positive radially symmetric ground state if either the coupling coefficient satisfies beta >Lambda (for an appropriate constant Lambda > 0) or if beta > 0 under appropriate conditions on the other parameters of the problem. We also prove that there exists a positive radially symmetric bound state if either 0 < beta is sufficiently small or if 0 < beta < Lambda under some appropriate conditions on the parameters. These results give a classification of positive solutions as well as the multiplicity of positive solutions. Furthermore, we study systems with more general power nonlinearities and systems with more than two nonlinear Schrödinger-Korteweg-de Vries equations. Our variational approach (working on the full energy functional without the L2-mass constraint) improves many previously known results and also allows us to show new results for some range of parameters not considered in the past.

keywords

  • variational methods; ground states; bound states; critical point theory; perturbation methods