Homoclinic Solutions for p(t)-Laplacian-Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian-Hamiltonian systems d/dt(vertical bar(u) over dot(t)vertical bar p(t)(-2)(u) over dot(t)) - a(t)vertical bar u(t)vertical bar(p(t)-2) u(t) + del W (t, u(t)) = 0, where , , with p(t) > 1, , and is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants such that for all and W(t, u) is of super-p(t) growth or sub-p(t) growth as , we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.