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We consider the non-linear eigenvalue equations characterizing $L^p$ into$L^q$ Sobolev embeddings of second order for Navier boundary conditions at bothends of a line segment. We give a complete description of the s-numbers and theextremal functions in the general case $(p,q)\in(1,\infty)^2$. Among otherresults, we show that these can be expressed in terms of those of related firstorder embeddings, if and only if $\frac{1}{p}+\frac{1}{q}=1$. Our findings shednew light on the surprising nature of higher order Sobolev spaces in the Banachspace setting.

The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings