Homoclinic solutions for some second order non-autonomous Hamiltonian systems without the globally superquadratic condition
From the abstract: This paper deals with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system \[ \ddot{q}+V_q(t,q)=f(t), \] where \(V\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R}), V(t,q)=-K(t,q)+W(t,q)\) is \(T\)-periodic in \(t\), \(f\) is aperiodic and belongs to \(L^2(\mathbb{R},\mathbb{R}^n)\). Under the assumptions that \(K\) satisfies the ``pinching'' condition \(b_1|q|^2\leq K(t,q)\leq b_2|q|^2\), \(W(t,q)\) is not globally superquadratic on \(q\) and some additionally reasonable assumptions, we give a new existence result to guarantee that the above system has a homoclinic solution \(q(t)\) emanating from \(0\). The homoclinic solution \(q(t)\) is obtained as a limit of \(2kT\)-periodic solutions of a sequence of the second order differential equations and these periodic solutions are obtained by the use of a standard version of the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.
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