A unified proof on existence of homoclinic orbits for some semilinear ordinary differential equations with periodic potentials

This paper gives a direct, short and unified proof of Rabinowitzʹs Theorem, Grossinho–Minhós–Tersianʹs Theorem and Tersian–Chaparovaʹs Theorems on existence of homoclinic orbits for second order periodic Hamiltonian systems, fourth and sixth order periodic ordinary differential equations, respectively, by Brezis–Nirenberg type Mountain Pass Lemma.