R.D. Benguria, I. Catto, J. Dolbeault, R. Monneau 
Oscillating minimizers of a fourth order problem invariant under scaling
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ABSTRACT.  By variational methods, we prove the inequality 
\[ 
\int_\R u''{}^2\,dx-\int_\R u''\,u^2\,dx\geq I\,\int_\R u^4\,dx 
\] 
for all $u\in L^4(\R)$ such that $u''\in L^2(\R)$ and for some constant $I\in (-9/64,-1/4)$. This inequality is connected to Lieb-Thirring type problems and has interesting scaling properties. The best constant is achieved by sign changing minimizers of a problem on periodic functions, but does not depend on the period. Moreover, we completely characterize the minimizers of the periodic problem.