Fractional Hamiltonian Systems with Locally Defined Potentials

Abstract

We study solutions of the nonperiodic fractional Hamiltonian systems

$${ - _t}D_\infty ^\alpha {(_{ - \infty }}D_\infty ^\alpha x(t)) - L(t)x(t) + \nabla W(t,x(t)) = 0,x \in {H^\alpha }(R,{R^N}),$$

where α ∈ (1/2, 1], t ∈ R, L(t) ∈ C(R,${R^{{N^2}}}$ ), and _−∞D_t^α t and tDα∞ are the respective left and right Liouville–Weyl fractional derivatives of order α on the whole axis R. Using a new symmetric mountain pass theorem established by Kajikia, we prove the existence of infinitely many solutions for this system in the case where the matrix L(t) is not necessarily coercive nor uniformly positive definite and W(t, x) is defined only locally near the coordinate origin x = 0. The proved theorems significantly generalize and improve previously obtained results. We also give several illustrative examples.