Singular endpoints in higher-order variational problems

The Euler-Lagrange equation on an open interval can be derived using compactly supported variations even when an endpoint is singular. A formal solution with singular endpoint data need not make the improper action finite. For example, $y=x+x^{-1}$ solves $(x^4{y}^{\prime \prime }{)}^{\prime \prime }=0$, but substituting into $x^4(y^{\prime \prime }{)}^2+4y^2{y}^{\prime }$ gives $4x^2+4-4x^{-4}$, whose integral down to zero diverges. Stationarity of a finite functional then requires an additional regularization or admissible-space specification.