Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system

This paper is concerned with the following nonperiodic Hamiltonian elliptic system { − Δ u + V ( x ) u = H v ( x , u , v ) − Δ v + V ( x ) v = H u ( x , u , v ) x ∈ R N , u ( x ) , v ( x ) → 0 as | x | → ∞ , where z = ( u , v ) : R N → R × R , N ≥ 3 . Assuming the potential V is nonperiodic and sign-changing, H ( x , z ) is nonperiodic in x and asymptotically quadratic in z = ( u , v ) , existence and multiplicity of solutions are obtained via a variational approach.