On the existence of bound and ground states for some coupled nonlinear Schrödinger–Korteweg–de Vries equations

Abstract

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 β > Λ {\beta>\Lambda} (for an appropriate constant Λ > 0 {\Lambda>0} ) or if β > 0 {\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 < β {0<\beta} is sufficiently small or if 0 < β < Λ {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 L 2 {L^{2}} -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.