Existence of multiple solutions for a class of second-order ordinary differential equations

By means of variational structure and $Z_2$ group index theory, we obtain multiple solutions for the second-order differential equation $$ frac{d}{dt}(p(t)frac{du}{dt})+q(t)u+f(t,u)=0 ,quad 0 less than t less than 1, $$ subject to one of the following two sets of boundary conditions: $$ u'(0) = u(1)+ u'(1)=0quadhbox{or}quad u(0)=u(1)=0,. $$