Multiple homoclinic solutions for p-Laplacian Hamiltonian systems with concave–convex nonlinearities

Abstract

AbstractThe multiplicity of homoclinic solutions is obtained for a class of thep-Laplacian Hamiltonian systems$\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))-a(t)|u(t)|^{p-2}u(t)+ \nabla W(t,u(t))=0$ddt(|u˙(t)|p−2u˙(t))−a(t)|u(t)|p−2u(t)+∇W(t,u(t))=0via variational methods, where$a(t)$a(t)is neither coercive nor bounded necessarily and$W(t,u)$W(t,u)is under new concave–convex conditions. Recent results in the literature are generalized even for$p=2$p=2.