We prove the existence and the multiplicity of positive solutions of the one-dimensional capillarity-type problem $$ -\left({u'}/{\sqrt{1+(u')^2}}\right)' = a(x) f(u), \quad u'(0)=0,\,\,u'(1)=0, $$ where $a\in L^1(0,1)$ changes sign and $f : [0,+\infty) \, \to [0,+\infty)$ is continuous and has a power-like behavior at the origin and at infinity.
Our approach is variational and relies on a regularization procedure that yields bounded variation solutions which are of class $W_\mathrm{loc}^{2,1}$, and hence classically satisfy the equation, on each open interval where the weight function $a$ has a constant sign.
