From the introduction: We establish the existence of solutions to a mixed boundary value problem of ordinary differential system with a \(p\)-Laplacian in the form \[ (\phi_p(u'))'+\nabla F(t,u)= 0,\qquad u(0)= u'(1)= 0, \] where \(\phi_p(x)=| x|^{p-2}x\) for \(x\in\mathbb{R}^n\) with \(|x|= (\sum^n_{i=1} x^2_i)^{{1\over 2}}\) and \(1<p<2\) by using the mountain pass theorem.