On the existence of homoclinic orbits for a second-order Hamiltonian system

Abstract

In this paper we look for homoclinic solutions of the system $$ \ddot q-a(t)\mid q\mid ^{p-2}q+W_q(t,q)=0 $$ where $p>2,$ $a(t)\rightarrow +\infty $ as $\mid q\mid \rightarrow +\infty $ and $W(t,\cdot )$ is even and quadratic or superquadratic at infinity and at the origin. Using a compact embedding between suitable weighted Sobolev spaces, we prove the existence of infinitely many homoclinic solutions of the problem.