Systems with coupling in \(\mathbb R^N\) for a class of noncoercive potentials (Q1422521)

The authors study the existence and multiplicity of solutions for the problem \[ -\Delta u+ a(x)u= F_u(x,u,v),\quad x\in\mathbb{R}^N, \] \[ -\Delta v+ b(x)v= F_v(x,u,v),\quad x\in\mathbb{R}^N, \] where \(N\geq 3\) and the potentials \(a\), \(b\) as well as \(F\) satisfy natural assumptions. They assume that the system is coupled and resonant. The existence of solution is proved under a critical growth condition on the nonlinearity.