A new proof of the existence of homoclinic orbits for a class of autonomous second order Hamiltonian systems in IRN

We consider the Hamiltonian system in R-N given by (u) double overdot + V'(u) = 0 where V : R-N --> R is a smooth potential having a non degenerate local maximum at 0 and we assume that there is an open bounded neighborhood Omega of 0 such that V(x) < V(0) for x is an element of Omega\{0}, V(x) = V(0) and V'(x) not equal 0 for x is an element of partial derivative Omega. Using a refined version of the mountain pass lemma [4], we give a further proof of the existence of a solution of u + V'(u) = 0, homoclinic to 0.