Solitary waves for coupled nonlinear elliptic system with nonhomogeneous nonlinearities

In present paper we consider a class of coupled elliptic system with nonhomogeneous nonlinearities. This type of system is related to the Raman amplification in a plasma. We make rigorous study and find the threshold conditions to guarantee the existence, nonexistence and multiplicity of nontrivial solutions for both two and three coupled system by using Morse theory, direct analysis methods and Krasnosel’skii–Rabinowitz global bifurcation theorem. Moreover, we study the asymptotical behavior of positive solutions, and prove some interesting phenomena for these solutions. Comparing to our previous works Wang and Shi (standing waves for weakly coupled Schrödinger equations with quadratic nonlinearities. Preprint, 2015) on the homogeneous case, we encounter some new challenges in proving the existence and multiplicity of nontrivial solutions. We overcome these difficult by combining the Mountain–Pass theorem in convex set and the Nehari constraint methods.