Positive solutions of a one-dimensional indefinite capillarity-type problem: a variational approach

We prove the existence and the multiplicity of positive solutions of the one-dimensional capillarity-type problem−(u′/1+(u′)2)′=a(x)f(u),u′(0)=0,u′(1)=0, where a∈L1(0,1) changes sign and f:[0,+∞)→[0,+∞) 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 Wloc2,1, and hence satisfy the equation pointwise almost everywhere, on each open interval where the weight function a has a constant sign.