Existence of Positive and Sign-Changing Solutions to a Coupled Elliptic System with Mixed Nonlinearity Growth

Abstract

In the present paper, we make a rigorous study of the solitary wave solutions to a coupled Schrodinger system with quadratic and cubic nonlinearity. This kind of system of Schrodinger equations arises from optics theory. First, the existence and nonexistence of nontrivial solutions, respectively, in focusing and defocusing cases are considered. Second, we prove the existence of multiple nontrivial solutions by using the Crandall–Rabinowitz local bifurcation theorems and calculate the exact Morse index of these solutions. Third, the continuous dependence on the parameter and asymptotic behavior of positive ground state solutions in the focusing case are also established. Particularly, from the mathematical point of view, we prove the behavior of positive solution coincides with the physical phenomena of Bang et al. (Opt Lett 22(22):1680–1682, 1997; Phys Rev E (3) 58(4):5057–5069, 1998). Finally, we prove the existence of sign-changing solutions.