Multiple bounded variation solutions for a prescribed mean curvature equation with Neumann boundary conditions

We prove the existence of multiple positive BV-solutions Neumann problem $$ \begin{cases} \displaystyle -\left(\frac{u'}{\sqrt{1+u'^2}}\right)'=a(x)f(u)\quad&\mbox{in }(0,1), u'(0)=u'(1)=0,& {cases} where $a(x) > 0$ and $f$ belongs to a class nonlinear functions whose prototype example is given by $f(u) = -\lambda u + u^p$, for $\lambda $p 1$. In particular, $f(0)=0$ has unique zero, denoted $u_0$. Solutions are distinguished number intersections (in generalized sense) with constant solution $u u_0$. further that solutions found have continuous energy we also give sufficient conditions on nonlinearity get classical solutions. The analysis performed using an approximation mean curvature operator shooting method.