Solving the simplified model of the Hartree-Fock Bogoliubov (HFB) equation in coordinate space with the correct asymptotic boundary conditions, the spin-orbit splitting, occupation probabilities, effective pair-gap, and spin-response function are studied for weakly bound p neutrons. As the binding energy of P3/2 neutrons becomes small or approaches zero, the spin-orbit splitting P3/2-P1/2 in the one-body potential drastically decreases and, at the same time, the effective pair-gap for the p neutrons becomes small, while the occupation probability of the P3/2 level decreases only slightly. Consequently, in that limit low-lying broader spin-response with almost constant amount of total strength appears moving toward very low excitation energies.