Ground and bound state solutions for a Schr\'odinger system with linear and nonlinear couplings in $\mathbb{R}^N$

We study the existence of ground and bound state solutions for a system of coupled Schr\"odinger equations with linear and nonlinear couplings in $\mathbb{R}^N$. By studying the limit system and using concentration compactness arguments, we prove the existence of ground and bound state solutions under suitable assumptions. Our results are new even for the limit system.