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

Abstract

Abstract The multiplicity of homoclinic solutions is obtained for a class of the p-Laplacian Hamiltonian systems d d t ( | u ˙ ( t ) | p − 2 u ˙ ( t ) ) − a ( t ) | u ( t ) | p − 2 u ( t ) + ∇ W ( t , u ( t ) ) = 0 $\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))-a(t)|u(t)|^{p-2}u(t)+ \nabla W(t,u(t))=0$ via 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 $p=2$ .

Keywords

• Homoclinic solutions
• p-Laplacian operator
• Variational methods