Existence and concentration of semiclassical solutions for Hamiltonian elliptic system (Q906812)

This paper is concerned with the system in \({\mathbb R}^N\): \[ -\epsilon^2 \Delta \psi+\epsilon b\cdot \nabla \psi+\psi+V(x)\varphi=K(x)f(|\eta|)\varphi \] \[ -\epsilon^2 \Delta \varphi+\epsilon b\cdot \nabla \varphi+\psi+V(x)\psi=K(x)f(|\eta|)\psi, \] where \(\eta=(\psi,\varphi)\), \(\epsilon>0\) is small, \(b\) is a constant vector, \(V\) is a sign-changing function with at least one global minimum, \(K\) has at least one global maximum. The main result of the paper establishes the existence, exponential decay and concentration of semiclassical ground state solutions of the above problem. The approach is variational, the solution is obtained as a critical point of the energy functional which does not satisfy the Palais-Smale condition.